Theorem axpre-sup 9990

Description: A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version with ordering on extended reals is axsup 10113. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 10014. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
axpre-sup	⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))

Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem axpre-sup

Dummy variables 𝑤 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elreal2 9953	. . . . . . 7 ⊢ (𝑥 ∈ ℝ ↔ ((1st ‘𝑥) ∈ R ∧ 𝑥 = ⟨(1st ‘𝑥), 0R⟩))
2	1	simplbi 476	. . . . . 6 ⊢ (𝑥 ∈ ℝ → (1st ‘𝑥) ∈ R)
3	2	adantl 482	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ) → (1st ‘𝑥) ∈ R)
4		fo1st 7188	. . . . . . . . . . . 12 ⊢ 1st :V–onto→V
5		fof 6115	. . . . . . . . . . . 12 ⊢ (1st :V–onto→V → 1st :V⟶V)
6		ffn 6045	. . . . . . . . . . . 12 ⊢ (1st :V⟶V → 1st Fn V)
7	4, 5, 6	mp2b 10	. . . . . . . . . . 11 ⊢ 1st Fn V
8		ssv 3625	. . . . . . . . . . 11 ⊢ 𝐴 ⊆ V
9		fvelimab 6253	. . . . . . . . . . 11 ⊢ ((1st Fn V ∧ 𝐴 ⊆ V) → (𝑤 ∈ (1st “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 (1st ‘𝑦) = 𝑤))
10	7, 8, 9	mp2an 708	. . . . . . . . . 10 ⊢ (𝑤 ∈ (1st “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 (1st ‘𝑦) = 𝑤)
11		r19.29 3072	. . . . . . . . . . . 12 ⊢ ((∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∃𝑦 ∈ 𝐴 (1st ‘𝑦) = 𝑤) → ∃𝑦 ∈ 𝐴 (𝑦 <ℝ 𝑥 ∧ (1st ‘𝑦) = 𝑤))
12		ssel2 3598	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ)
13		ltresr2 9962	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 <ℝ 𝑥 ↔ (1st ‘𝑦) <R (1st ‘𝑥)))
14		breq1 4656	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑦) = 𝑤 → ((1st ‘𝑦) <R (1st ‘𝑥) ↔ 𝑤 <R (1st ‘𝑥)))
15	13, 14	sylan9bb 736	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ (1st ‘𝑦) = 𝑤) → (𝑦 <ℝ 𝑥 ↔ 𝑤 <R (1st ‘𝑥)))
16	15	biimpd 219	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ (1st ‘𝑦) = 𝑤) → (𝑦 <ℝ 𝑥 → 𝑤 <R (1st ‘𝑥)))
17	16	exp31 630	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ → (𝑥 ∈ ℝ → ((1st ‘𝑦) = 𝑤 → (𝑦 <ℝ 𝑥 → 𝑤 <R (1st ‘𝑥)))))
18	12, 17	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ ℝ → ((1st ‘𝑦) = 𝑤 → (𝑦 <ℝ 𝑥 → 𝑤 <R (1st ‘𝑥)))))
19	18	imp4b 613	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ ℝ) → (((1st ‘𝑦) = 𝑤 ∧ 𝑦 <ℝ 𝑥) → 𝑤 <R (1st ‘𝑥)))
20	19	ancomsd 470	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ ℝ) → ((𝑦 <ℝ 𝑥 ∧ (1st ‘𝑦) = 𝑤) → 𝑤 <R (1st ‘𝑥)))
21	20	an32s 846	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → ((𝑦 <ℝ 𝑥 ∧ (1st ‘𝑦) = 𝑤) → 𝑤 <R (1st ‘𝑥)))
22	21	rexlimdva 3031	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∃𝑦 ∈ 𝐴 (𝑦 <ℝ 𝑥 ∧ (1st ‘𝑦) = 𝑤) → 𝑤 <R (1st ‘𝑥)))
23	11, 22	syl5 34	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → ((∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∃𝑦 ∈ 𝐴 (1st ‘𝑦) = 𝑤) → 𝑤 <R (1st ‘𝑥)))
24	23	expd 452	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 → (∃𝑦 ∈ 𝐴 (1st ‘𝑦) = 𝑤 → 𝑤 <R (1st ‘𝑥))))
25	10, 24	syl7bi 245	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 → (𝑤 ∈ (1st “ 𝐴) → 𝑤 <R (1st ‘𝑥))))
26	25	impr 649	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) → (𝑤 ∈ (1st “ 𝐴) → 𝑤 <R (1st ‘𝑥)))
27	26	adantlr 751	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) → (𝑤 ∈ (1st “ 𝐴) → 𝑤 <R (1st ‘𝑥)))
28	27	ralrimiv 2965	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) → ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R (1st ‘𝑥))
29	28	expr 643	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 → ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R (1st ‘𝑥)))
30		breq2 4657	. . . . . . 7 ⊢ (𝑣 = (1st ‘𝑥) → (𝑤 <R 𝑣 ↔ 𝑤 <R (1st ‘𝑥)))
31	30	ralbidv 2986	. . . . . 6 ⊢ (𝑣 = (1st ‘𝑥) → (∀𝑤 ∈ (1st “ 𝐴)𝑤 <R 𝑣 ↔ ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R (1st ‘𝑥)))
32	31	rspcev 3309	. . . . 5 ⊢ (((1st ‘𝑥) ∈ R ∧ ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R (1st ‘𝑥)) → ∃𝑣 ∈ R ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R 𝑣)
33	3, 29, 32	syl6an 568	. . . 4 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 → ∃𝑣 ∈ R ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R 𝑣))
34	33	rexlimdva 3031	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 → ∃𝑣 ∈ R ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R 𝑣))
35		n0 3931	. . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐴)
36		fnfvima 6496	. . . . . . . . 9 ⊢ ((1st Fn V ∧ 𝐴 ⊆ V ∧ 𝑦 ∈ 𝐴) → (1st ‘𝑦) ∈ (1st “ 𝐴))
37	7, 8, 36	mp3an12 1414	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (1st ‘𝑦) ∈ (1st “ 𝐴))
38		ne0i 3921	. . . . . . . 8 ⊢ ((1st ‘𝑦) ∈ (1st “ 𝐴) → (1st “ 𝐴) ≠ ∅)
39	37, 38	syl 17	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → (1st “ 𝐴) ≠ ∅)
40	39	exlimiv 1858	. . . . . 6 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (1st “ 𝐴) ≠ ∅)
41	35, 40	sylbi 207	. . . . 5 ⊢ (𝐴 ≠ ∅ → (1st “ 𝐴) ≠ ∅)
42		supsr 9933	. . . . . 6 ⊢ (((1st “ 𝐴) ≠ ∅ ∧ ∃𝑣 ∈ R ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R 𝑣) → ∃𝑣 ∈ R (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 ∧ ∀𝑤 ∈ R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st “ 𝐴)𝑤 <R 𝑢)))
43	42	ex 450	. . . . 5 ⊢ ((1st “ 𝐴) ≠ ∅ → (∃𝑣 ∈ R ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R 𝑣 → ∃𝑣 ∈ R (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 ∧ ∀𝑤 ∈ R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st “ 𝐴)𝑤 <R 𝑢))))
44	41, 43	syl 17	. . . 4 ⊢ (𝐴 ≠ ∅ → (∃𝑣 ∈ R ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R 𝑣 → ∃𝑣 ∈ R (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 ∧ ∀𝑤 ∈ R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st “ 𝐴)𝑤 <R 𝑢))))
45	44	adantl 482	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (∃𝑣 ∈ R ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R 𝑣 → ∃𝑣 ∈ R (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 ∧ ∀𝑤 ∈ R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st “ 𝐴)𝑤 <R 𝑢))))
46		breq2 4657	. . . . . . . . . . . 12 ⊢ (𝑤 = (1st ‘𝑦) → (𝑣 <R 𝑤 ↔ 𝑣 <R (1st ‘𝑦)))
47	46	notbid 308	. . . . . . . . . . 11 ⊢ (𝑤 = (1st ‘𝑦) → (¬ 𝑣 <R 𝑤 ↔ ¬ 𝑣 <R (1st ‘𝑦)))
48	47	rspccv 3306	. . . . . . . . . 10 ⊢ (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 → ((1st ‘𝑦) ∈ (1st “ 𝐴) → ¬ 𝑣 <R (1st ‘𝑦)))
49	37, 48	syl5com 31	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 → ¬ 𝑣 <R (1st ‘𝑦)))
50	49	adantl 482	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 → ¬ 𝑣 <R (1st ‘𝑦)))
51		elreal2 9953	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ ↔ ((1st ‘𝑦) ∈ R ∧ 𝑦 = ⟨(1st ‘𝑦), 0R⟩))
52	51	simprbi 480	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 𝑦 = ⟨(1st ‘𝑦), 0R⟩)
53	52	breq2d 4665	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → (⟨𝑣, 0R⟩ <ℝ 𝑦 ↔ ⟨𝑣, 0R⟩ <ℝ ⟨(1st ‘𝑦), 0R⟩))
54		ltresr 9961	. . . . . . . . . . 11 ⊢ (⟨𝑣, 0R⟩ <ℝ ⟨(1st ‘𝑦), 0R⟩ ↔ 𝑣 <R (1st ‘𝑦))
55	53, 54	syl6bb 276	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → (⟨𝑣, 0R⟩ <ℝ 𝑦 ↔ 𝑣 <R (1st ‘𝑦)))
56	12, 55	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → (⟨𝑣, 0R⟩ <ℝ 𝑦 ↔ 𝑣 <R (1st ‘𝑦)))
57	56	notbid 308	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → (¬ ⟨𝑣, 0R⟩ <ℝ 𝑦 ↔ ¬ 𝑣 <R (1st ‘𝑦)))
58	50, 57	sylibrd 249	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 → ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦))
59	58	ralrimdva 2969	. . . . . 6 ⊢ (𝐴 ⊆ ℝ → (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 → ∀𝑦 ∈ 𝐴 ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦))
60	59	ad2antrr 762	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣 ∈ R) → (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 → ∀𝑦 ∈ 𝐴 ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦))
61	52	breq1d 4663	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ → (𝑦 <ℝ ⟨𝑣, 0R⟩ ↔ ⟨(1st ‘𝑦), 0R⟩ <ℝ ⟨𝑣, 0R⟩))
62		ltresr 9961	. . . . . . . . . . . . . 14 ⊢ (⟨(1st ‘𝑦), 0R⟩ <ℝ ⟨𝑣, 0R⟩ ↔ (1st ‘𝑦) <R 𝑣)
63	61, 62	syl6bb 276	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → (𝑦 <ℝ ⟨𝑣, 0R⟩ ↔ (1st ‘𝑦) <R 𝑣))
64	51	simplbi 476	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ → (1st ‘𝑦) ∈ R)
65		breq1 4656	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (1st ‘𝑦) → (𝑤 <R 𝑣 ↔ (1st ‘𝑦) <R 𝑣))
66		breq1 4656	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (1st ‘𝑦) → (𝑤 <R 𝑢 ↔ (1st ‘𝑦) <R 𝑢))
67	66	rexbidv 3052	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (1st ‘𝑦) → (∃𝑢 ∈ (1st “ 𝐴)𝑤 <R 𝑢 ↔ ∃𝑢 ∈ (1st “ 𝐴)(1st ‘𝑦) <R 𝑢))
68	65, 67	imbi12d 334	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (1st ‘𝑦) → ((𝑤 <R 𝑣 → ∃𝑢 ∈ (1st “ 𝐴)𝑤 <R 𝑢) ↔ ((1st ‘𝑦) <R 𝑣 → ∃𝑢 ∈ (1st “ 𝐴)(1st ‘𝑦) <R 𝑢)))
69	68	rspccv 3306	. . . . . . . . . . . . . . 15 ⊢ (∀𝑤 ∈ R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st “ 𝐴)𝑤 <R 𝑢) → ((1st ‘𝑦) ∈ R → ((1st ‘𝑦) <R 𝑣 → ∃𝑢 ∈ (1st “ 𝐴)(1st ‘𝑦) <R 𝑢)))
70	64, 69	syl5 34	. . . . . . . . . . . . . 14 ⊢ (∀𝑤 ∈ R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st “ 𝐴)𝑤 <R 𝑢) → (𝑦 ∈ ℝ → ((1st ‘𝑦) <R 𝑣 → ∃𝑢 ∈ (1st “ 𝐴)(1st ‘𝑦) <R 𝑢)))
71	70	com3l 89	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → ((1st ‘𝑦) <R 𝑣 → (∀𝑤 ∈ R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st “ 𝐴)𝑤 <R 𝑢) → ∃𝑢 ∈ (1st “ 𝐴)(1st ‘𝑦) <R 𝑢)))
72	63, 71	sylbid 230	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → (𝑦 <ℝ ⟨𝑣, 0R⟩ → (∀𝑤 ∈ R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st “ 𝐴)𝑤 <R 𝑢) → ∃𝑢 ∈ (1st “ 𝐴)(1st ‘𝑦) <R 𝑢)))
73	72	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (𝑦 <ℝ ⟨𝑣, 0R⟩ → (∀𝑤 ∈ R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st “ 𝐴)𝑤 <R 𝑢) → ∃𝑢 ∈ (1st “ 𝐴)(1st ‘𝑦) <R 𝑢)))
74		fvelimab 6253	. . . . . . . . . . . . . . . 16 ⊢ ((1st Fn V ∧ 𝐴 ⊆ V) → (𝑢 ∈ (1st “ 𝐴) ↔ ∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑢))
75	7, 8, 74	mp2an 708	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (1st “ 𝐴) ↔ ∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑢)
76		ssel2 3598	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ)
77		ltresr2 9962	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑦 <ℝ 𝑧 ↔ (1st ‘𝑦) <R (1st ‘𝑧)))
78	76, 77	sylan2 491	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴)) → (𝑦 <ℝ 𝑧 ↔ (1st ‘𝑦) <R (1st ‘𝑧)))
79		breq2 4657	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1st ‘𝑧) = 𝑢 → ((1st ‘𝑦) <R (1st ‘𝑧) ↔ (1st ‘𝑦) <R 𝑢))
80	78, 79	sylan9bb 736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴)) ∧ (1st ‘𝑧) = 𝑢) → (𝑦 <ℝ 𝑧 ↔ (1st ‘𝑦) <R 𝑢))
81	80	exbiri 652	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴)) → ((1st ‘𝑧) = 𝑢 → ((1st ‘𝑦) <R 𝑢 → 𝑦 <ℝ 𝑧)))
82	81	expr 643	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (𝑧 ∈ 𝐴 → ((1st ‘𝑧) = 𝑢 → ((1st ‘𝑦) <R 𝑢 → 𝑦 <ℝ 𝑧))))
83	82	com4r 94	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘𝑦) <R 𝑢 → ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (𝑧 ∈ 𝐴 → ((1st ‘𝑧) = 𝑢 → 𝑦 <ℝ 𝑧))))
84	83	imp 445	. . . . . . . . . . . . . . . 16 ⊢ (((1st ‘𝑦) <R 𝑢 ∧ (𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ)) → (𝑧 ∈ 𝐴 → ((1st ‘𝑧) = 𝑢 → 𝑦 <ℝ 𝑧)))
85	84	reximdvai 3015	. . . . . . . . . . . . . . 15 ⊢ (((1st ‘𝑦) <R 𝑢 ∧ (𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ)) → (∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑢 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))
86	75, 85	syl5bi 232	. . . . . . . . . . . . . 14 ⊢ (((1st ‘𝑦) <R 𝑢 ∧ (𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ)) → (𝑢 ∈ (1st “ 𝐴) → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))
87	86	expcom 451	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → ((1st ‘𝑦) <R 𝑢 → (𝑢 ∈ (1st “ 𝐴) → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
88	87	com23 86	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (𝑢 ∈ (1st “ 𝐴) → ((1st ‘𝑦) <R 𝑢 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
89	88	rexlimdv 3030	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (∃𝑢 ∈ (1st “ 𝐴)(1st ‘𝑦) <R 𝑢 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))
90	73, 89	syl6d 75	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (𝑦 <ℝ ⟨𝑣, 0R⟩ → (∀𝑤 ∈ R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st “ 𝐴)𝑤 <R 𝑢) → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
91	90	com23 86	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (∀𝑤 ∈ R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st “ 𝐴)𝑤 <R 𝑢) → (𝑦 <ℝ ⟨𝑣, 0R⟩ → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
92	91	ex 450	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (𝐴 ⊆ ℝ → (∀𝑤 ∈ R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st “ 𝐴)𝑤 <R 𝑢) → (𝑦 <ℝ ⟨𝑣, 0R⟩ → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))))
93	92	com3l 89	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ → (∀𝑤 ∈ R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st “ 𝐴)𝑤 <R 𝑢) → (𝑦 ∈ ℝ → (𝑦 <ℝ ⟨𝑣, 0R⟩ → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))))
94	93	ad2antrr 762	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣 ∈ R) → (∀𝑤 ∈ R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st “ 𝐴)𝑤 <R 𝑢) → (𝑦 ∈ ℝ → (𝑦 <ℝ ⟨𝑣, 0R⟩ → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))))
95	94	ralrimdv 2968	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣 ∈ R) → (∀𝑤 ∈ R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st “ 𝐴)𝑤 <R 𝑢) → ∀𝑦 ∈ ℝ (𝑦 <ℝ ⟨𝑣, 0R⟩ → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
96		opelreal 9951	. . . . . . . 8 ⊢ (⟨𝑣, 0R⟩ ∈ ℝ ↔ 𝑣 ∈ R)
97	96	biimpri 218	. . . . . . 7 ⊢ (𝑣 ∈ R → ⟨𝑣, 0R⟩ ∈ ℝ)
98	97	adantl 482	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣 ∈ R) → ⟨𝑣, 0R⟩ ∈ ℝ)
99		breq1 4656	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝑣, 0R⟩ → (𝑥 <ℝ 𝑦 ↔ ⟨𝑣, 0R⟩ <ℝ 𝑦))
100	99	notbid 308	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑣, 0R⟩ → (¬ 𝑥 <ℝ 𝑦 ↔ ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦))
101	100	ralbidv 2986	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑣, 0R⟩ → (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦))
102		breq2 4657	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝑣, 0R⟩ → (𝑦 <ℝ 𝑥 ↔ 𝑦 <ℝ ⟨𝑣, 0R⟩))
103	102	imbi1d 331	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑣, 0R⟩ → ((𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧) ↔ (𝑦 <ℝ ⟨𝑣, 0R⟩ → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
104	103	ralbidv 2986	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑣, 0R⟩ → (∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧) ↔ ∀𝑦 ∈ ℝ (𝑦 <ℝ ⟨𝑣, 0R⟩ → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
105	101, 104	anbi12d 747	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑣, 0R⟩ → ((∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)) ↔ (∀𝑦 ∈ 𝐴 ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ ⟨𝑣, 0R⟩ → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))))
106	105	rspcev 3309	. . . . . . 7 ⊢ ((⟨𝑣, 0R⟩ ∈ ℝ ∧ (∀𝑦 ∈ 𝐴 ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ ⟨𝑣, 0R⟩ → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
107	106	ex 450	. . . . . 6 ⊢ (⟨𝑣, 0R⟩ ∈ ℝ → ((∀𝑦 ∈ 𝐴 ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ ⟨𝑣, 0R⟩ → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))))
108	98, 107	syl 17	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣 ∈ R) → ((∀𝑦 ∈ 𝐴 ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ ⟨𝑣, 0R⟩ → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))))
109	60, 95, 108	syl2and 500	. . . 4 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣 ∈ R) → ((∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 ∧ ∀𝑤 ∈ R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st “ 𝐴)𝑤 <R 𝑢)) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))))
110	109	rexlimdva 3031	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (∃𝑣 ∈ R (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 ∧ ∀𝑤 ∈ R (𝑤 <R 𝑣 → ∃𝑢 ∈ (1st “ 𝐴)𝑤 <R 𝑢)) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))))
111	34, 45, 110	3syld 60	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))))
112	111	3impia 1261	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  ⟨cop 4183   class class class wbr 4653   “ cima 5117   Fn wfn 5883  ⟶wf 5884  –onto→wfo 5886  ‘cfv 5888  1^st c1st 7166  Rcnr 9687  0_Rc0r 9688   <_R cltr 9693  ℝcr 9935   <_ℝ cltrr 9940

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-1p 9804  df-plp 9805  df-mp 9806  df-ltp 9807  df-enr 9877  df-nr 9878  df-plr 9879  df-mr 9880  df-ltr 9881  df-0r 9882  df-1r 9883  df-m1r 9884  df-r 9946  df-lt 9949

This theorem is referenced by: (None)