Theorem axpre-sup 11164

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 11289. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 11188. (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 11127	. . . . . . 7 ⊢ (𝑥 ∈ ℝ ↔ ((1st ‘𝑥) ∈ R ∧ 𝑥 = ⟨(1st ‘𝑥), 0R⟩))
2	1	simplbi 499	. . . . . 6 ⊢ (𝑥 ∈ ℝ → (1st ‘𝑥) ∈ R)
3	2	adantl 483	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ) → (1st ‘𝑥) ∈ R)
4		fo1st 7995	. . . . . . . . . . . 12 ⊢ 1st :V–onto→V
5		fof 6806	. . . . . . . . . . . 12 ⊢ (1st :V–onto→V → 1st :V⟶V)
6		ffn 6718	. . . . . . . . . . . 12 ⊢ (1st :V⟶V → 1st Fn V)
7	4, 5, 6	mp2b 10	. . . . . . . . . . 11 ⊢ 1st Fn V
8		ssv 4007	. . . . . . . . . . 11 ⊢ 𝐴 ⊆ V
9		fvelimab 6965	. . . . . . . . . . 11 ⊢ ((1st Fn V ∧ 𝐴 ⊆ V) → (𝑤 ∈ (1st “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 (1st ‘𝑦) = 𝑤))
10	7, 8, 9	mp2an 691	. . . . . . . . . 10 ⊢ (𝑤 ∈ (1st “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 (1st ‘𝑦) = 𝑤)
11		r19.29 3115	. . . . . . . . . . . 12 ⊢ ((∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∃𝑦 ∈ 𝐴 (1st ‘𝑦) = 𝑤) → ∃𝑦 ∈ 𝐴 (𝑦 <ℝ 𝑥 ∧ (1st ‘𝑦) = 𝑤))
12		ssel2 3978	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ)
13		ltresr2 11136	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 <ℝ 𝑥 ↔ (1st ‘𝑦) <R (1st ‘𝑥)))
14		breq1 5152	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑦) = 𝑤 → ((1st ‘𝑦) <R (1st ‘𝑥) ↔ 𝑤 <R (1st ‘𝑥)))
15	13, 14	sylan9bb 511	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ (1st ‘𝑦) = 𝑤) → (𝑦 <ℝ 𝑥 ↔ 𝑤 <R (1st ‘𝑥)))
16	15	biimpd 228	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ (1st ‘𝑦) = 𝑤) → (𝑦 <ℝ 𝑥 → 𝑤 <R (1st ‘𝑥)))
17	16	exp31 421	. . . . . . . . . . . . . . . . 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 423	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ ℝ) → (((1st ‘𝑦) = 𝑤 ∧ 𝑦 <ℝ 𝑥) → 𝑤 <R (1st ‘𝑥)))
20	19	ancomsd 467	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ ℝ) → ((𝑦 <ℝ 𝑥 ∧ (1st ‘𝑦) = 𝑤) → 𝑤 <R (1st ‘𝑥)))
21	20	an32s 651	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → ((𝑦 <ℝ 𝑥 ∧ (1st ‘𝑦) = 𝑤) → 𝑤 <R (1st ‘𝑥)))
22	21	rexlimdva 3156	. . . . . . . . . . . 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 417	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 → (∃𝑦 ∈ 𝐴 (1st ‘𝑦) = 𝑤 → 𝑤 <R (1st ‘𝑥))))
25	10, 24	syl7bi 255	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 → (𝑤 ∈ (1st “ 𝐴) → 𝑤 <R (1st ‘𝑥))))
26	25	impr 456	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) → (𝑤 ∈ (1st “ 𝐴) → 𝑤 <R (1st ‘𝑥)))
27	26	adantlr 714	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) → (𝑤 ∈ (1st “ 𝐴) → 𝑤 <R (1st ‘𝑥)))
28	27	ralrimiv 3146	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) → ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R (1st ‘𝑥))
29	28	expr 458	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 → ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R (1st ‘𝑥)))
30		brralrspcev 5209	. . . . 5 ⊢ (((1st ‘𝑥) ∈ R ∧ ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R (1st ‘𝑥)) → ∃𝑣 ∈ R ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R 𝑣)
31	3, 29, 30	syl6an 683	. . . 4 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 → ∃𝑣 ∈ R ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R 𝑣))
32	31	rexlimdva 3156	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 → ∃𝑣 ∈ R ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R 𝑣))
33		n0 4347	. . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐴)
34		fnfvima 7235	. . . . . . . . 9 ⊢ ((1st Fn V ∧ 𝐴 ⊆ V ∧ 𝑦 ∈ 𝐴) → (1st ‘𝑦) ∈ (1st “ 𝐴))
35	7, 8, 34	mp3an12 1452	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (1st ‘𝑦) ∈ (1st “ 𝐴))
36	35	ne0d 4336	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → (1st “ 𝐴) ≠ ∅)
37	36	exlimiv 1934	. . . . . 6 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (1st “ 𝐴) ≠ ∅)
38	33, 37	sylbi 216	. . . . 5 ⊢ (𝐴 ≠ ∅ → (1st “ 𝐴) ≠ ∅)
39		supsr 11107	. . . . . 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)))
40	39	ex 414	. . . . 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))))
41	38, 40	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))))
42	41	adantl 483	. . 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))))
43		breq2 5153	. . . . . . . . . . . 12 ⊢ (𝑤 = (1st ‘𝑦) → (𝑣 <R 𝑤 ↔ 𝑣 <R (1st ‘𝑦)))
44	43	notbid 318	. . . . . . . . . . 11 ⊢ (𝑤 = (1st ‘𝑦) → (¬ 𝑣 <R 𝑤 ↔ ¬ 𝑣 <R (1st ‘𝑦)))
45	44	rspccv 3610	. . . . . . . . . 10 ⊢ (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 → ((1st ‘𝑦) ∈ (1st “ 𝐴) → ¬ 𝑣 <R (1st ‘𝑦)))
46	35, 45	syl5com 31	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 → ¬ 𝑣 <R (1st ‘𝑦)))
47	46	adantl 483	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 → ¬ 𝑣 <R (1st ‘𝑦)))
48		elreal2 11127	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ ↔ ((1st ‘𝑦) ∈ R ∧ 𝑦 = ⟨(1st ‘𝑦), 0R⟩))
49	48	simprbi 498	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 𝑦 = ⟨(1st ‘𝑦), 0R⟩)
50	49	breq2d 5161	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → (⟨𝑣, 0R⟩ <ℝ 𝑦 ↔ ⟨𝑣, 0R⟩ <ℝ ⟨(1st ‘𝑦), 0R⟩))
51		ltresr 11135	. . . . . . . . . . 11 ⊢ (⟨𝑣, 0R⟩ <ℝ ⟨(1st ‘𝑦), 0R⟩ ↔ 𝑣 <R (1st ‘𝑦))
52	50, 51	bitrdi 287	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → (⟨𝑣, 0R⟩ <ℝ 𝑦 ↔ 𝑣 <R (1st ‘𝑦)))
53	12, 52	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → (⟨𝑣, 0R⟩ <ℝ 𝑦 ↔ 𝑣 <R (1st ‘𝑦)))
54	53	notbid 318	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → (¬ ⟨𝑣, 0R⟩ <ℝ 𝑦 ↔ ¬ 𝑣 <R (1st ‘𝑦)))
55	47, 54	sylibrd 259	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 → ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦))
56	55	ralrimdva 3155	. . . . . 6 ⊢ (𝐴 ⊆ ℝ → (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 → ∀𝑦 ∈ 𝐴 ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦))
57	56	ad2antrr 725	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣 ∈ R) → (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 → ∀𝑦 ∈ 𝐴 ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦))
58	49	breq1d 5159	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ → (𝑦 <ℝ ⟨𝑣, 0R⟩ ↔ ⟨(1st ‘𝑦), 0R⟩ <ℝ ⟨𝑣, 0R⟩))
59		ltresr 11135	. . . . . . . . . . . . . 14 ⊢ (⟨(1st ‘𝑦), 0R⟩ <ℝ ⟨𝑣, 0R⟩ ↔ (1st ‘𝑦) <R 𝑣)
60	58, 59	bitrdi 287	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → (𝑦 <ℝ ⟨𝑣, 0R⟩ ↔ (1st ‘𝑦) <R 𝑣))
61	48	simplbi 499	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ → (1st ‘𝑦) ∈ R)
62		breq1 5152	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (1st ‘𝑦) → (𝑤 <R 𝑣 ↔ (1st ‘𝑦) <R 𝑣))
63		breq1 5152	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (1st ‘𝑦) → (𝑤 <R 𝑢 ↔ (1st ‘𝑦) <R 𝑢))
64	63	rexbidv 3179	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (1st ‘𝑦) → (∃𝑢 ∈ (1st “ 𝐴)𝑤 <R 𝑢 ↔ ∃𝑢 ∈ (1st “ 𝐴)(1st ‘𝑦) <R 𝑢))
65	62, 64	imbi12d 345	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (1st ‘𝑦) → ((𝑤 <R 𝑣 → ∃𝑢 ∈ (1st “ 𝐴)𝑤 <R 𝑢) ↔ ((1st ‘𝑦) <R 𝑣 → ∃𝑢 ∈ (1st “ 𝐴)(1st ‘𝑦) <R 𝑢)))
66	65	rspccv 3610	. . . . . . . . . . . . . . 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)))
67	61, 66	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)))
68	67	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)))
69	60, 68	sylbid 239	. . . . . . . . . . . 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)))
70	69	adantr 482	. . . . . . . . . . 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)))
71		fvelimab 6965	. . . . . . . . . . . . . . . 16 ⊢ ((1st Fn V ∧ 𝐴 ⊆ V) → (𝑢 ∈ (1st “ 𝐴) ↔ ∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑢))
72	7, 8, 71	mp2an 691	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (1st “ 𝐴) ↔ ∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑢)
73		ssel2 3978	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ)
74		ltresr2 11136	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑦 <ℝ 𝑧 ↔ (1st ‘𝑦) <R (1st ‘𝑧)))
75	73, 74	sylan2 594	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴)) → (𝑦 <ℝ 𝑧 ↔ (1st ‘𝑦) <R (1st ‘𝑧)))
76		breq2 5153	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1st ‘𝑧) = 𝑢 → ((1st ‘𝑦) <R (1st ‘𝑧) ↔ (1st ‘𝑦) <R 𝑢))
77	75, 76	sylan9bb 511	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴)) ∧ (1st ‘𝑧) = 𝑢) → (𝑦 <ℝ 𝑧 ↔ (1st ‘𝑦) <R 𝑢))
78	77	exbiri 810	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴)) → ((1st ‘𝑧) = 𝑢 → ((1st ‘𝑦) <R 𝑢 → 𝑦 <ℝ 𝑧)))
79	78	expr 458	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (𝑧 ∈ 𝐴 → ((1st ‘𝑧) = 𝑢 → ((1st ‘𝑦) <R 𝑢 → 𝑦 <ℝ 𝑧))))
80	79	com4r 94	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘𝑦) <R 𝑢 → ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (𝑧 ∈ 𝐴 → ((1st ‘𝑧) = 𝑢 → 𝑦 <ℝ 𝑧))))
81	80	imp 408	. . . . . . . . . . . . . . . 16 ⊢ (((1st ‘𝑦) <R 𝑢 ∧ (𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ)) → (𝑧 ∈ 𝐴 → ((1st ‘𝑧) = 𝑢 → 𝑦 <ℝ 𝑧)))
82	81	reximdvai 3166	. . . . . . . . . . . . . . 15 ⊢ (((1st ‘𝑦) <R 𝑢 ∧ (𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ)) → (∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑢 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))
83	72, 82	biimtrid 241	. . . . . . . . . . . . . 14 ⊢ (((1st ‘𝑦) <R 𝑢 ∧ (𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ)) → (𝑢 ∈ (1st “ 𝐴) → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))
84	83	expcom 415	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → ((1st ‘𝑦) <R 𝑢 → (𝑢 ∈ (1st “ 𝐴) → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
85	84	com23 86	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (𝑢 ∈ (1st “ 𝐴) → ((1st ‘𝑦) <R 𝑢 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
86	85	rexlimdv 3154	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (∃𝑢 ∈ (1st “ 𝐴)(1st ‘𝑦) <R 𝑢 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))
87	70, 86	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)))
88	87	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)))
89	88	ex 414	. . . . . . . 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))))
90	89	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))))
91	90	ad2antrr 725	. . . . . 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))))
92	91	ralrimdv 3153	. . . . 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)))
93		opelreal 11125	. . . . . . . 8 ⊢ (⟨𝑣, 0R⟩ ∈ ℝ ↔ 𝑣 ∈ R)
94	93	biimpri 227	. . . . . . 7 ⊢ (𝑣 ∈ R → ⟨𝑣, 0R⟩ ∈ ℝ)
95	94	adantl 483	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣 ∈ R) → ⟨𝑣, 0R⟩ ∈ ℝ)
96		breq1 5152	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝑣, 0R⟩ → (𝑥 <ℝ 𝑦 ↔ ⟨𝑣, 0R⟩ <ℝ 𝑦))
97	96	notbid 318	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑣, 0R⟩ → (¬ 𝑥 <ℝ 𝑦 ↔ ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦))
98	97	ralbidv 3178	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑣, 0R⟩ → (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦))
99		breq2 5153	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝑣, 0R⟩ → (𝑦 <ℝ 𝑥 ↔ 𝑦 <ℝ ⟨𝑣, 0R⟩))
100	99	imbi1d 342	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑣, 0R⟩ → ((𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧) ↔ (𝑦 <ℝ ⟨𝑣, 0R⟩ → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
101	100	ralbidv 3178	. . . . . . . . 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)))
102	98, 101	anbi12d 632	. . . . . . . 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))))
103	102	rspcev 3613	. . . . . . 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)))
104	103	ex 414	. . . . . 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))))
105	95, 104	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))))
106	57, 92, 105	syl2and 609	. . . 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))))
107	106	rexlimdva 3156	. . 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))))
108	32, 42, 107	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))))
109	108	3impia 1118	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 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  ∃wex 1782   ∈ wcel 2107   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  Vcvv 3475   ⊆ wss 3949  ∅c0 4323  ⟨cop 4635   class class class wbr 5149   “ cima 5680   Fn wfn 6539  ⟶wf 6540  –onto→wfo 6542  ‘cfv 6544  1^st c1st 7973  Rcnr 10860  0_Rc0r 10861   <_R cltr 10866  ℝcr 11109   <_ℝ cltrr 11114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-omul 8471  df-er 8703  df-ec 8705  df-qs 8709  df-ni 10867  df-pli 10868  df-mi 10869  df-lti 10870  df-plpq 10903  df-mpq 10904  df-ltpq 10905  df-enq 10906  df-nq 10907  df-erq 10908  df-plq 10909  df-mq 10910  df-1nq 10911  df-rq 10912  df-ltnq 10913  df-np 10976  df-1p 10977  df-plp 10978  df-mp 10979  df-ltp 10980  df-enr 11050  df-nr 11051  df-plr 11052  df-mr 11053  df-ltr 11054  df-0r 11055  df-1r 11056  df-m1r 11057  df-r 11120  df-lt 11123

This theorem is referenced by: (None)