Theorem slerec 31923

Description: A comparison law for surreals considered as cuts of sets of surreals. In Conway's treatment, this is the definition of less than or equal. (Contributed by Scott Fenton, 11-Dec-2021.)

Assertion

Ref	Expression
slerec	⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 ≤s 𝑌 ↔ (∀𝑑 ∈ 𝐷 𝑋 <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑌)))

Distinct variable groups:   𝐴,𝑎,𝑑   𝐵,𝑎,𝑑   𝐶,𝑎,𝑑   𝐷,𝑎,𝑑   𝑋,𝑎,𝑑   𝑌,𝑎,𝑑

Proof of Theorem slerec

Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		scutcut 31912	. . . . . . . . 9 ⊢ (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
2	1	simp1d 1073	. . . . . . . 8 ⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ No )
3	2	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑 ∈ 𝐷) → (𝐴 |s 𝐵) ∈ No )
4		scutcut 31912	. . . . . . . . 9 ⊢ (𝐶 <<s 𝐷 → ((𝐶 |s 𝐷) ∈ No ∧ 𝐶 <<s {(𝐶 |s 𝐷)} ∧ {(𝐶 |s 𝐷)} <<s 𝐷))
5	4	simp1d 1073	. . . . . . . 8 ⊢ (𝐶 <<s 𝐷 → (𝐶 |s 𝐷) ∈ No )
6	5	ad3antlr 767	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑 ∈ 𝐷) → (𝐶 |s 𝐷) ∈ No )
7		ssltss2 31904	. . . . . . . . 9 ⊢ (𝐶 <<s 𝐷 → 𝐷 ⊆ No )
8	7	ad2antlr 763	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → 𝐷 ⊆ No )
9	8	sselda 3603	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ No )
10		simplr 792	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑 ∈ 𝐷) → (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷))
11	4	simp3d 1075	. . . . . . . . . . 11 ⊢ (𝐶 <<s 𝐷 → {(𝐶 |s 𝐷)} <<s 𝐷)
12	11	ad2antlr 763	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → {(𝐶 |s 𝐷)} <<s 𝐷)
13		ssltsep 31905	. . . . . . . . . 10 ⊢ ({(𝐶 |s 𝐷)} <<s 𝐷 → ∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑑 ∈ 𝐷 𝑎 <s 𝑑)
14	12, 13	syl 17	. . . . . . . . 9 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑑 ∈ 𝐷 𝑎 <s 𝑑)
15		ovex 6678	. . . . . . . . . 10 ⊢ (𝐶 |s 𝐷) ∈ V
16		breq1 4656	. . . . . . . . . . 11 ⊢ (𝑎 = (𝐶 |s 𝐷) → (𝑎 <s 𝑑 ↔ (𝐶 |s 𝐷) <s 𝑑))
17	16	ralbidv 2986	. . . . . . . . . 10 ⊢ (𝑎 = (𝐶 |s 𝐷) → (∀𝑑 ∈ 𝐷 𝑎 <s 𝑑 ↔ ∀𝑑 ∈ 𝐷 (𝐶 |s 𝐷) <s 𝑑))
18	15, 17	ralsn 4222	. . . . . . . . 9 ⊢ (∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑑 ∈ 𝐷 𝑎 <s 𝑑 ↔ ∀𝑑 ∈ 𝐷 (𝐶 |s 𝐷) <s 𝑑)
19	14, 18	sylib 208	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑑 ∈ 𝐷 (𝐶 |s 𝐷) <s 𝑑)
20	19	r19.21bi 2932	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑 ∈ 𝐷) → (𝐶 |s 𝐷) <s 𝑑)
21	3, 6, 9, 10, 20	slelttrd 31886	. . . . . 6 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑 ∈ 𝐷) → (𝐴 |s 𝐵) <s 𝑑)
22	21	ralrimiva 2966	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑)
23		ssltss1 31903	. . . . . . . . . 10 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
24	23	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) → 𝐴 ⊆ No )
25	24	adantr 481	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → 𝐴 ⊆ No )
26	25	sselda 3603	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ No )
27	2	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎 ∈ 𝐴) → (𝐴 |s 𝐵) ∈ No )
28	5	ad3antlr 767	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎 ∈ 𝐴) → (𝐶 |s 𝐷) ∈ No )
29	1	simp2d 1074	. . . . . . . . . . . 12 ⊢ (𝐴 <<s 𝐵 → 𝐴 <<s {(𝐴 |s 𝐵)})
30	29	adantr 481	. . . . . . . . . . 11 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) → 𝐴 <<s {(𝐴 |s 𝐵)})
31	30	adantr 481	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → 𝐴 <<s {(𝐴 |s 𝐵)})
32		ssltsep 31905	. . . . . . . . . 10 ⊢ (𝐴 <<s {(𝐴 |s 𝐵)} → ∀𝑎 ∈ 𝐴 ∀𝑑 ∈ {(𝐴 |s 𝐵)}𝑎 <s 𝑑)
33	31, 32	syl 17	. . . . . . . . 9 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑎 ∈ 𝐴 ∀𝑑 ∈ {(𝐴 |s 𝐵)}𝑎 <s 𝑑)
34	33	r19.21bi 2932	. . . . . . . 8 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎 ∈ 𝐴) → ∀𝑑 ∈ {(𝐴 |s 𝐵)}𝑎 <s 𝑑)
35		ovex 6678	. . . . . . . . 9 ⊢ (𝐴 |s 𝐵) ∈ V
36		breq2 4657	. . . . . . . . 9 ⊢ (𝑑 = (𝐴 |s 𝐵) → (𝑎 <s 𝑑 ↔ 𝑎 <s (𝐴 |s 𝐵)))
37	35, 36	ralsn 4222	. . . . . . . 8 ⊢ (∀𝑑 ∈ {(𝐴 |s 𝐵)}𝑎 <s 𝑑 ↔ 𝑎 <s (𝐴 |s 𝐵))
38	34, 37	sylib 208	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝑎 <s (𝐴 |s 𝐵))
39		simplr 792	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎 ∈ 𝐴) → (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷))
40	26, 27, 28, 38, 39	sltletrd 31885	. . . . . 6 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝑎 <s (𝐶 |s 𝐷))
41	40	ralrimiva 2966	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))
42	22, 41	jca 554	. . . 4 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷)))
43		bdayelon 31892	. . . . . . 7 ⊢ ( bday ‘(𝐴 |s 𝐵)) ∈ On
44	43	onordi 5832	. . . . . 6 ⊢ Ord ( bday ‘(𝐴 |s 𝐵))
45		ordn2lp 5743	. . . . . 6 ⊢ (Ord ( bday ‘(𝐴 |s 𝐵)) → ¬ (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘(𝐶 |s 𝐷)) ∧ ( bday ‘(𝐶 |s 𝐷)) ∈ ( bday ‘(𝐴 |s 𝐵))))
46	44, 45	ax-mp 5	. . . . 5 ⊢ ¬ (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘(𝐶 |s 𝐷)) ∧ ( bday ‘(𝐶 |s 𝐷)) ∈ ( bday ‘(𝐴 |s 𝐵)))
47	5	ad2antlr 763	. . . . . . 7 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) → (𝐶 |s 𝐷) ∈ No )
48	2	adantr 481	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) → (𝐴 |s 𝐵) ∈ No )
49	48	adantr 481	. . . . . . 7 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) → (𝐴 |s 𝐵) ∈ No )
50		sltnle 31878	. . . . . . 7 ⊢ (((𝐶 |s 𝐷) ∈ No ∧ (𝐴 |s 𝐵) ∈ No ) → ((𝐶 |s 𝐷) <s (𝐴 |s 𝐵) ↔ ¬ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)))
51	47, 49, 50	syl2anc 693	. . . . . 6 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) → ((𝐶 |s 𝐷) <s (𝐴 |s 𝐵) ↔ ¬ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)))
52	5	ad3antlr 767	. . . . . . . . 9 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐶 |s 𝐷) ∈ No )
53		ssltex1 31901	. . . . . . . . . . . 12 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
54	53	ad3antrrr 766	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐴 ∈ V)
55		snex 4908	. . . . . . . . . . 11 ⊢ {(𝐶 |s 𝐷)} ∈ V
56	54, 55	jctir 561	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐴 ∈ V ∧ {(𝐶 |s 𝐷)} ∈ V))
57	23	ad3antrrr 766	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐴 ⊆ No )
58	52	snssd 4340	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → {(𝐶 |s 𝐷)} ⊆ No )
59		simplrr 801	. . . . . . . . . . . 12 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))
60		breq2 4657	. . . . . . . . . . . . . 14 ⊢ (𝑑 = (𝐶 |s 𝐷) → (𝑎 <s 𝑑 ↔ 𝑎 <s (𝐶 |s 𝐷)))
61	15, 60	ralsn 4222	. . . . . . . . . . . . 13 ⊢ (∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑎 <s 𝑑 ↔ 𝑎 <s (𝐶 |s 𝐷))
62	61	ralbii 2980	. . . . . . . . . . . 12 ⊢ (∀𝑎 ∈ 𝐴 ∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑎 <s 𝑑 ↔ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))
63	59, 62	sylibr 224	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑎 ∈ 𝐴 ∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑎 <s 𝑑)
64	57, 58, 63	3jca 1242	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐴 ⊆ No ∧ {(𝐶 |s 𝐷)} ⊆ No ∧ ∀𝑎 ∈ 𝐴 ∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑎 <s 𝑑))
65		brsslt 31900	. . . . . . . . . 10 ⊢ (𝐴 <<s {(𝐶 |s 𝐷)} ↔ ((𝐴 ∈ V ∧ {(𝐶 |s 𝐷)} ∈ V) ∧ (𝐴 ⊆ No ∧ {(𝐶 |s 𝐷)} ⊆ No ∧ ∀𝑎 ∈ 𝐴 ∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑎 <s 𝑑)))
66	56, 64, 65	sylanbrc 698	. . . . . . . . 9 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐴 <<s {(𝐶 |s 𝐷)})
67		ssltex2 31902	. . . . . . . . . . . 12 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
68	67	ad3antrrr 766	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐵 ∈ V)
69	68, 55	jctil 560	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ({(𝐶 |s 𝐷)} ∈ V ∧ 𝐵 ∈ V))
70		ssltss2 31904	. . . . . . . . . . . 12 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
71	70	ad3antrrr 766	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐵 ⊆ No )
72	52	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑏 ∈ 𝐵) → (𝐶 |s 𝐷) ∈ No )
73	48	ad3antrrr 766	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑏 ∈ 𝐵) → (𝐴 |s 𝐵) ∈ No )
74	71	sselda 3603	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ No )
75		simplr 792	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑏 ∈ 𝐵) → (𝐶 |s 𝐷) <s (𝐴 |s 𝐵))
76	1	simp3d 1075	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 <<s 𝐵 → {(𝐴 |s 𝐵)} <<s 𝐵)
77	76	ad3antrrr 766	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → {(𝐴 |s 𝐵)} <<s 𝐵)
78		ssltsep 31905	. . . . . . . . . . . . . . . . 17 ⊢ ({(𝐴 |s 𝐵)} <<s 𝐵 → ∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑏 ∈ 𝐵 𝑎 <s 𝑏)
79	77, 78	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑏 ∈ 𝐵 𝑎 <s 𝑏)
80		breq1 4656	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (𝐴 |s 𝐵) → (𝑎 <s 𝑏 ↔ (𝐴 |s 𝐵) <s 𝑏))
81	80	ralbidv 2986	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (𝐴 |s 𝐵) → (∀𝑏 ∈ 𝐵 𝑎 <s 𝑏 ↔ ∀𝑏 ∈ 𝐵 (𝐴 |s 𝐵) <s 𝑏))
82	35, 81	ralsn 4222	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑏 ∈ 𝐵 𝑎 <s 𝑏 ↔ ∀𝑏 ∈ 𝐵 (𝐴 |s 𝐵) <s 𝑏)
83	79, 82	sylib 208	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑏 ∈ 𝐵 (𝐴 |s 𝐵) <s 𝑏)
84	83	r19.21bi 2932	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑏 ∈ 𝐵) → (𝐴 |s 𝐵) <s 𝑏)
85	72, 73, 74, 75, 84	slttrd 31884	. . . . . . . . . . . . 13 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑏 ∈ 𝐵) → (𝐶 |s 𝐷) <s 𝑏)
86	85	ralrimiva 2966	. . . . . . . . . . . 12 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑏 ∈ 𝐵 (𝐶 |s 𝐷) <s 𝑏)
87		breq1 4656	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝐶 |s 𝐷) → (𝑎 <s 𝑏 ↔ (𝐶 |s 𝐷) <s 𝑏))
88	87	ralbidv 2986	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝐶 |s 𝐷) → (∀𝑏 ∈ 𝐵 𝑎 <s 𝑏 ↔ ∀𝑏 ∈ 𝐵 (𝐶 |s 𝐷) <s 𝑏))
89	15, 88	ralsn 4222	. . . . . . . . . . . 12 ⊢ (∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑏 ∈ 𝐵 𝑎 <s 𝑏 ↔ ∀𝑏 ∈ 𝐵 (𝐶 |s 𝐷) <s 𝑏)
90	86, 89	sylibr 224	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑏 ∈ 𝐵 𝑎 <s 𝑏)
91	58, 71, 90	3jca 1242	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ({(𝐶 |s 𝐷)} ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑏 ∈ 𝐵 𝑎 <s 𝑏))
92		brsslt 31900	. . . . . . . . . 10 ⊢ ({(𝐶 |s 𝐷)} <<s 𝐵 ↔ (({(𝐶 |s 𝐷)} ∈ V ∧ 𝐵 ∈ V) ∧ ({(𝐶 |s 𝐷)} ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑏 ∈ 𝐵 𝑎 <s 𝑏)))
93	69, 91, 92	sylanbrc 698	. . . . . . . . 9 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → {(𝐶 |s 𝐷)} <<s 𝐵)
94		sltirr 31871	. . . . . . . . . . . . . 14 ⊢ ((𝐴 |s 𝐵) ∈ No → ¬ (𝐴 |s 𝐵) <s (𝐴 |s 𝐵))
95	49, 94	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) → ¬ (𝐴 |s 𝐵) <s (𝐴 |s 𝐵))
96		breq1 4656	. . . . . . . . . . . . . 14 ⊢ ((𝐴 |s 𝐵) = (𝐶 |s 𝐷) → ((𝐴 |s 𝐵) <s (𝐴 |s 𝐵) ↔ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)))
97	96	notbid 308	. . . . . . . . . . . . 13 ⊢ ((𝐴 |s 𝐵) = (𝐶 |s 𝐷) → (¬ (𝐴 |s 𝐵) <s (𝐴 |s 𝐵) ↔ ¬ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)))
98	95, 97	syl5ibcom 235	. . . . . . . . . . . 12 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) → ((𝐴 |s 𝐵) = (𝐶 |s 𝐷) → ¬ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)))
99	98	necon2ad 2809	. . . . . . . . . . 11 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) → ((𝐶 |s 𝐷) <s (𝐴 |s 𝐵) → (𝐴 |s 𝐵) ≠ (𝐶 |s 𝐷)))
100	99	imp 445	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐴 |s 𝐵) ≠ (𝐶 |s 𝐷))
101	100	necomd 2849	. . . . . . . . 9 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐶 |s 𝐷) ≠ (𝐴 |s 𝐵))
102		scutbdaylt 31922	. . . . . . . . 9 ⊢ (((𝐶 |s 𝐷) ∈ No ∧ (𝐴 <<s {(𝐶 |s 𝐷)} ∧ {(𝐶 |s 𝐷)} <<s 𝐵) ∧ (𝐶 |s 𝐷) ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘(𝐶 |s 𝐷)))
103	52, 66, 93, 101, 102	syl121anc 1331	. . . . . . . 8 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘(𝐶 |s 𝐷)))
104	2	ad3antrrr 766	. . . . . . . . 9 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐴 |s 𝐵) ∈ No )
105		ssltex1 31901	. . . . . . . . . . . 12 ⊢ (𝐶 <<s 𝐷 → 𝐶 ∈ V)
106	105	ad3antlr 767	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐶 ∈ V)
107		snex 4908	. . . . . . . . . . 11 ⊢ {(𝐴 |s 𝐵)} ∈ V
108	106, 107	jctir 561	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐶 ∈ V ∧ {(𝐴 |s 𝐵)} ∈ V))
109		ssltss1 31903	. . . . . . . . . . . 12 ⊢ (𝐶 <<s 𝐷 → 𝐶 ⊆ No )
110	109	ad3antlr 767	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐶 ⊆ No )
111	104	snssd 4340	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → {(𝐴 |s 𝐵)} ⊆ No )
112	110	sselda 3603	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐 ∈ 𝐶) → 𝑐 ∈ No )
113	52	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐 ∈ 𝐶) → (𝐶 |s 𝐷) ∈ No )
114	48	ad3antrrr 766	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐 ∈ 𝐶) → (𝐴 |s 𝐵) ∈ No )
115	4	simp2d 1074	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 <<s 𝐷 → 𝐶 <<s {(𝐶 |s 𝐷)})
116	115	ad3antlr 767	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐶 <<s {(𝐶 |s 𝐷)})
117		ssltsep 31905	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 <<s {(𝐶 |s 𝐷)} → ∀𝑐 ∈ 𝐶 ∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑐 <s 𝑑)
118	116, 117	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑐 ∈ 𝐶 ∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑐 <s 𝑑)
119	118	r19.21bi 2932	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐 ∈ 𝐶) → ∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑐 <s 𝑑)
120		breq2 4657	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = (𝐶 |s 𝐷) → (𝑐 <s 𝑑 ↔ 𝑐 <s (𝐶 |s 𝐷)))
121	15, 120	ralsn 4222	. . . . . . . . . . . . . . 15 ⊢ (∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑐 <s 𝑑 ↔ 𝑐 <s (𝐶 |s 𝐷))
122	119, 121	sylib 208	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐 ∈ 𝐶) → 𝑐 <s (𝐶 |s 𝐷))
123		simplr 792	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐 ∈ 𝐶) → (𝐶 |s 𝐷) <s (𝐴 |s 𝐵))
124	112, 113, 114, 122, 123	slttrd 31884	. . . . . . . . . . . . 13 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐 ∈ 𝐶) → 𝑐 <s (𝐴 |s 𝐵))
125		breq2 4657	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝐴 |s 𝐵) → (𝑐 <s 𝑎 ↔ 𝑐 <s (𝐴 |s 𝐵)))
126	35, 125	ralsn 4222	. . . . . . . . . . . . 13 ⊢ (∀𝑎 ∈ {(𝐴 |s 𝐵)}𝑐 <s 𝑎 ↔ 𝑐 <s (𝐴 |s 𝐵))
127	124, 126	sylibr 224	. . . . . . . . . . . 12 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐 ∈ 𝐶) → ∀𝑎 ∈ {(𝐴 |s 𝐵)}𝑐 <s 𝑎)
128	127	ralrimiva 2966	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑐 ∈ 𝐶 ∀𝑎 ∈ {(𝐴 |s 𝐵)}𝑐 <s 𝑎)
129	110, 111, 128	3jca 1242	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐶 ⊆ No ∧ {(𝐴 |s 𝐵)} ⊆ No ∧ ∀𝑐 ∈ 𝐶 ∀𝑎 ∈ {(𝐴 |s 𝐵)}𝑐 <s 𝑎))
130		brsslt 31900	. . . . . . . . . 10 ⊢ (𝐶 <<s {(𝐴 |s 𝐵)} ↔ ((𝐶 ∈ V ∧ {(𝐴 |s 𝐵)} ∈ V) ∧ (𝐶 ⊆ No ∧ {(𝐴 |s 𝐵)} ⊆ No ∧ ∀𝑐 ∈ 𝐶 ∀𝑎 ∈ {(𝐴 |s 𝐵)}𝑐 <s 𝑎)))
131	108, 129, 130	sylanbrc 698	. . . . . . . . 9 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐶 <<s {(𝐴 |s 𝐵)})
132		ssltex2 31902	. . . . . . . . . . . 12 ⊢ (𝐶 <<s 𝐷 → 𝐷 ∈ V)
133	132	ad3antlr 767	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐷 ∈ V)
134	133, 107	jctil 560	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ({(𝐴 |s 𝐵)} ∈ V ∧ 𝐷 ∈ V))
135	7	ad3antlr 767	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐷 ⊆ No )
136		simplrl 800	. . . . . . . . . . . 12 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑)
137		breq1 4656	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝐴 |s 𝐵) → (𝑎 <s 𝑑 ↔ (𝐴 |s 𝐵) <s 𝑑))
138	137	ralbidv 2986	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝐴 |s 𝐵) → (∀𝑑 ∈ 𝐷 𝑎 <s 𝑑 ↔ ∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑))
139	35, 138	ralsn 4222	. . . . . . . . . . . 12 ⊢ (∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑑 ∈ 𝐷 𝑎 <s 𝑑 ↔ ∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑)
140	136, 139	sylibr 224	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑑 ∈ 𝐷 𝑎 <s 𝑑)
141	111, 135, 140	3jca 1242	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ({(𝐴 |s 𝐵)} ⊆ No ∧ 𝐷 ⊆ No ∧ ∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑑 ∈ 𝐷 𝑎 <s 𝑑))
142		brsslt 31900	. . . . . . . . . 10 ⊢ ({(𝐴 |s 𝐵)} <<s 𝐷 ↔ (({(𝐴 |s 𝐵)} ∈ V ∧ 𝐷 ∈ V) ∧ ({(𝐴 |s 𝐵)} ⊆ No ∧ 𝐷 ⊆ No ∧ ∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑑 ∈ 𝐷 𝑎 <s 𝑑)))
143	134, 141, 142	sylanbrc 698	. . . . . . . . 9 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → {(𝐴 |s 𝐵)} <<s 𝐷)
144		scutbdaylt 31922	. . . . . . . . 9 ⊢ (((𝐴 |s 𝐵) ∈ No ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ (𝐴 |s 𝐵) ≠ (𝐶 |s 𝐷)) → ( bday ‘(𝐶 |s 𝐷)) ∈ ( bday ‘(𝐴 |s 𝐵)))
145	104, 131, 143, 100, 144	syl121anc 1331	. . . . . . . 8 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ( bday ‘(𝐶 |s 𝐷)) ∈ ( bday ‘(𝐴 |s 𝐵)))
146	103, 145	jca 554	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘(𝐶 |s 𝐷)) ∧ ( bday ‘(𝐶 |s 𝐷)) ∈ ( bday ‘(𝐴 |s 𝐵))))
147	146	ex 450	. . . . . 6 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) → ((𝐶 |s 𝐷) <s (𝐴 |s 𝐵) → (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘(𝐶 |s 𝐷)) ∧ ( bday ‘(𝐶 |s 𝐷)) ∈ ( bday ‘(𝐴 |s 𝐵)))))
148	51, 147	sylbird 250	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) → (¬ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷) → (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘(𝐶 |s 𝐷)) ∧ ( bday ‘(𝐶 |s 𝐷)) ∈ ( bday ‘(𝐴 |s 𝐵)))))
149	46, 148	mt3i 141	. . . 4 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) → (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷))
150	42, 149	impbida 877	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) → ((𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷) ↔ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))))
151		breq12 4658	. . . 4 ⊢ ((𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷)) → (𝑋 ≤s 𝑌 ↔ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)))
152		breq1 4656	. . . . . 6 ⊢ (𝑋 = (𝐴 |s 𝐵) → (𝑋 <s 𝑑 ↔ (𝐴 |s 𝐵) <s 𝑑))
153	152	ralbidv 2986	. . . . 5 ⊢ (𝑋 = (𝐴 |s 𝐵) → (∀𝑑 ∈ 𝐷 𝑋 <s 𝑑 ↔ ∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑))
154		breq2 4657	. . . . . 6 ⊢ (𝑌 = (𝐶 |s 𝐷) → (𝑎 <s 𝑌 ↔ 𝑎 <s (𝐶 |s 𝐷)))
155	154	ralbidv 2986	. . . . 5 ⊢ (𝑌 = (𝐶 |s 𝐷) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑌 ↔ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷)))
156	153, 155	bi2anan9 917	. . . 4 ⊢ ((𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷)) → ((∀𝑑 ∈ 𝐷 𝑋 <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑌) ↔ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))))
157	151, 156	bibi12d 335	. . 3 ⊢ ((𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷)) → ((𝑋 ≤s 𝑌 ↔ (∀𝑑 ∈ 𝐷 𝑋 <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑌)) ↔ ((𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷) ↔ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷)))))
158	150, 157	syl5ibr 236	. 2 ⊢ ((𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷)) → ((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) → (𝑋 ≤s 𝑌 ↔ (∀𝑑 ∈ 𝐷 𝑋 <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑌))))
159	158	impcom 446	1 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 ≤s 𝑌 ↔ (∀𝑑 ∈ 𝐷 𝑋 <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑌)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  {csn 4177   class class class wbr 4653  Ord word 5722  ‘cfv 5888  (class class class)co 6650   No csur 31793   <s cslt 31794   bday cbday 31795   ≤s csle 31869   <<s csslt 31896   |s cscut 31898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1o 7560  df-2o 7561  df-no 31796  df-slt 31797  df-bday 31798  df-sle 31870  df-sslt 31897  df-scut 31899

This theorem is referenced by:  sltrec  31924