Theorem wsuclemOLD 31774

Description: Obsolete version of wsuclem 31773 as of 10-Oct-2021. (Contributed by Scott Fenton, 15-Jun-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
wsuclem.1	⊢ (𝜑 → 𝑅 We 𝐴)
wsuclem.2	⊢ (𝜑 → 𝑅 Se 𝐴)
wsuclem.3	⊢ (𝜑 → 𝑋 ∈ 𝑉)
wsuclem.4	⊢ (𝜑 → ∃𝑤 ∈ 𝐴 𝑋𝑅𝑤)

Assertion

Ref	Expression
wsuclemOLD	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑦◡𝑅𝑧)))

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧,𝑤   𝜑,𝑥,𝑦   𝑥,𝑅,𝑦,𝑧,𝑤   𝑥,𝑋,𝑦,𝑧,𝑤

Allowed substitution hints:   𝜑(𝑧,𝑤)   𝑉(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem wsuclemOLD

Step	Hyp	Ref	Expression
1		wsuclem.1	. . 3 ⊢ (𝜑 → 𝑅 We 𝐴)
2		wsuclem.2	. . 3 ⊢ (𝜑 → 𝑅 Se 𝐴)
3		predss 5687	. . . 4 ⊢ Pred(◡𝑅, 𝐴, 𝑋) ⊆ 𝐴
4	3	a1i 11	. . 3 ⊢ (𝜑 → Pred(◡𝑅, 𝐴, 𝑋) ⊆ 𝐴)
5		wsuclem.3	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
6		dfpred3g 5691	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → Pred(◡𝑅, 𝐴, 𝑋) = {𝑤 ∈ 𝐴 ∣ 𝑤◡𝑅𝑋})
7	5, 6	syl 17	. . . 4 ⊢ (𝜑 → Pred(◡𝑅, 𝐴, 𝑋) = {𝑤 ∈ 𝐴 ∣ 𝑤◡𝑅𝑋})
8		elex 3212	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ V)
9	5, 8	syl 17	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ V)
10		wsuclem.4	. . . . 5 ⊢ (𝜑 → ∃𝑤 ∈ 𝐴 𝑋𝑅𝑤)
11		rabn0 3958	. . . . . . 7 ⊢ ({𝑤 ∈ 𝐴 ∣ 𝑤◡𝑅𝑋} ≠ ∅ ↔ ∃𝑤 ∈ 𝐴 𝑤◡𝑅𝑋)
12		brcnvg 5303	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝐴 ∧ 𝑋 ∈ V) → (𝑤◡𝑅𝑋 ↔ 𝑋𝑅𝑤))
13	12	ancoms 469	. . . . . . . 8 ⊢ ((𝑋 ∈ V ∧ 𝑤 ∈ 𝐴) → (𝑤◡𝑅𝑋 ↔ 𝑋𝑅𝑤))
14	13	rexbidva 3049	. . . . . . 7 ⊢ (𝑋 ∈ V → (∃𝑤 ∈ 𝐴 𝑤◡𝑅𝑋 ↔ ∃𝑤 ∈ 𝐴 𝑋𝑅𝑤))
15	11, 14	syl5bb 272	. . . . . 6 ⊢ (𝑋 ∈ V → ({𝑤 ∈ 𝐴 ∣ 𝑤◡𝑅𝑋} ≠ ∅ ↔ ∃𝑤 ∈ 𝐴 𝑋𝑅𝑤))
16	15	biimpar 502	. . . . 5 ⊢ ((𝑋 ∈ V ∧ ∃𝑤 ∈ 𝐴 𝑋𝑅𝑤) → {𝑤 ∈ 𝐴 ∣ 𝑤◡𝑅𝑋} ≠ ∅)
17	9, 10, 16	syl2anc 693	. . . 4 ⊢ (𝜑 → {𝑤 ∈ 𝐴 ∣ 𝑤◡𝑅𝑋} ≠ ∅)
18	7, 17	eqnetrd 2861	. . 3 ⊢ (𝜑 → Pred(◡𝑅, 𝐴, 𝑋) ≠ ∅)
19		tz6.26 5711	. . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (Pred(◡𝑅, 𝐴, 𝑋) ⊆ 𝐴 ∧ Pred(◡𝑅, 𝐴, 𝑋) ≠ ∅)) → ∃𝑥 ∈ Pred (◡𝑅, 𝐴, 𝑋)Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅)
20	1, 2, 4, 18, 19	syl22anc 1327	. 2 ⊢ (𝜑 → ∃𝑥 ∈ Pred (◡𝑅, 𝐴, 𝑋)Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅)
21		dfpred3g 5691	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → Pred(◡𝑅, 𝐴, 𝑋) = {𝑦 ∈ 𝐴 ∣ 𝑦◡𝑅𝑋})
22	5, 21	syl 17	. . . 4 ⊢ (𝜑 → Pred(◡𝑅, 𝐴, 𝑋) = {𝑦 ∈ 𝐴 ∣ 𝑦◡𝑅𝑋})
23	22	rexeqdv 3145	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ Pred (◡𝑅, 𝐴, 𝑋)Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅ ↔ ∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝑦◡𝑅𝑋}Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅))
24		breq1 4656	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦◡𝑅𝑋 ↔ 𝑥◡𝑅𝑋))
25	24	rexrab 3370	. . . 4 ⊢ (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝑦◡𝑅𝑋}Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅ ↔ ∃𝑥 ∈ 𝐴 (𝑥◡𝑅𝑋 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅))
26		noel 3919	. . . . . . . . . . . . 13 ⊢ ¬ 𝑦 ∈ ∅
27		simp2r 1088	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅) ∧ 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋)) → Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅)
28	27	eleq2d 2687	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅) ∧ 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋)) → (𝑦 ∈ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) ↔ 𝑦 ∈ ∅))
29	26, 28	mtbiri 317	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅) ∧ 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋)) → ¬ 𝑦 ∈ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥))
30		vex 3203	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
31	30	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅) ∧ 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋)) → 𝑥 ∈ V)
32		simp3 1063	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅) ∧ 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋)) → 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋))
33		elpredg 5694	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋)) → (𝑦 ∈ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) ↔ 𝑦𝑅𝑥))
34	31, 32, 33	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅) ∧ 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋)) → (𝑦 ∈ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) ↔ 𝑦𝑅𝑥))
35	29, 34	mtbid 314	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅) ∧ 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋)) → ¬ 𝑦𝑅𝑥)
36		vex 3203	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
37	30, 36	brcnv 5305	. . . . . . . . . . 11 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
38	35, 37	sylnibr 319	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅) ∧ 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋)) → ¬ 𝑥◡𝑅𝑦)
39	38	3expa 1265	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅)) ∧ 𝑦 ∈ Pred(◡𝑅, 𝐴, 𝑋)) → ¬ 𝑥◡𝑅𝑦)
40	39	ralrimiva 2966	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅)) → ∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑥◡𝑅𝑦)
41	40	expr 643	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅ → ∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑥◡𝑅𝑦))
42		simp1rl 1126	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑦◡𝑅𝑥) → 𝑥 ∈ 𝐴)
43		simp1rr 1127	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑦◡𝑅𝑥) → 𝑥◡𝑅𝑋)
44		simp1l 1085	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑦◡𝑅𝑥) → 𝜑)
45	44, 5	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑦◡𝑅𝑥) → 𝑋 ∈ 𝑉)
46	30	elpred 5693	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝑉 → (𝑥 ∈ Pred(◡𝑅, 𝐴, 𝑋) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)))
47	45, 46	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑦◡𝑅𝑥) → (𝑥 ∈ Pred(◡𝑅, 𝐴, 𝑋) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)))
48	42, 43, 47	mpbir2and 957	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑦◡𝑅𝑥) → 𝑥 ∈ Pred(◡𝑅, 𝐴, 𝑋))
49		simp3 1063	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑦◡𝑅𝑥) → 𝑦◡𝑅𝑥)
50		breq2 4657	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑦◡𝑅𝑧 ↔ 𝑦◡𝑅𝑥))
51	50	rspcev 3309	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ Pred(◡𝑅, 𝐴, 𝑋) ∧ 𝑦◡𝑅𝑥) → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑦◡𝑅𝑧)
52	48, 49, 51	syl2anc 693	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑦◡𝑅𝑥) → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑦◡𝑅𝑧)
53	52	3expia 1267	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) ∧ 𝑦 ∈ 𝐴) → (𝑦◡𝑅𝑥 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑦◡𝑅𝑧))
54	53	ralrimiva 2966	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥◡𝑅𝑋)) → ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑦◡𝑅𝑧))
55	54	expr 643	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥◡𝑅𝑋 → ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑦◡𝑅𝑧)))
56	41, 55	anim12d 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅ ∧ 𝑥◡𝑅𝑋) → (∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑦◡𝑅𝑧))))
57	56	ancomsd 470	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥◡𝑅𝑋 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅) → (∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑦◡𝑅𝑧))))
58	57	reximdva 3017	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 (𝑥◡𝑅𝑋 ∧ Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑦◡𝑅𝑧))))
59	25, 58	syl5bi 232	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝑦◡𝑅𝑋}Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅ → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑦◡𝑅𝑧))))
60	23, 59	sylbid 230	. 2 ⊢ (𝜑 → (∃𝑥 ∈ Pred (◡𝑅, 𝐴, 𝑋)Pred(𝑅, Pred(◡𝑅, 𝐴, 𝑋), 𝑥) = ∅ → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑦◡𝑅𝑧))))
61	20, 60	mpd 15	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑦◡𝑅𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ⊆ wss 3574  ∅c0 3915   class class class wbr 4653   Se wse 5071   We wwe 5072  ^◡ccnv 5113  Predcpred 5679

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680

This theorem is referenced by: (None)