Theorem axpre-sup 11166

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 11291. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 11190. (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 11129	. . . . . . 7 ⊢ (𝑥 ∈ ℝ ↔ ((1st ‘𝑥) ∈ R ∧ 𝑥 = ⟨(1st ‘𝑥), 0R⟩))
2	1	simplbi 498	. . . . . 6 ⊢ (𝑥 ∈ ℝ → (1st ‘𝑥) ∈ R)
3	2	adantl 482	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ) → (1st ‘𝑥) ∈ R)
4		fo1st 7997	. . . . . . . . . . . 12 ⊢ 1st :V–onto→V
5		fof 6805	. . . . . . . . . . . 12 ⊢ (1st :V–onto→V → 1st :V⟶V)
6		ffn 6717	. . . . . . . . . . . 12 ⊢ (1st :V⟶V → 1st Fn V)
7	4, 5, 6	mp2b 10	. . . . . . . . . . 11 ⊢ 1st Fn V
8		ssv 4006	. . . . . . . . . . 11 ⊢ 𝐴 ⊆ V
9		fvelimab 6964	. . . . . . . . . . 11 ⊢ ((1st Fn V ∧ 𝐴 ⊆ V) → (𝑤 ∈ (1st “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 (1st ‘𝑦) = 𝑤))
10	7, 8, 9	mp2an 690	. . . . . . . . . 10 ⊢ (𝑤 ∈ (1st “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 (1st ‘𝑦) = 𝑤)
11		r19.29 3114	. . . . . . . . . . . 12 ⊢ ((∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∃𝑦 ∈ 𝐴 (1st ‘𝑦) = 𝑤) → ∃𝑦 ∈ 𝐴 (𝑦 <ℝ 𝑥 ∧ (1st ‘𝑦) = 𝑤))
12		ssel2 3977	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ)
13		ltresr2 11138	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 <ℝ 𝑥 ↔ (1st ‘𝑦) <R (1st ‘𝑥)))
14		breq1 5151	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑦) = 𝑤 → ((1st ‘𝑦) <R (1st ‘𝑥) ↔ 𝑤 <R (1st ‘𝑥)))
15	13, 14	sylan9bb 510	. . . . . . . . . . . . . . . . . . 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 420	. . . . . . . . . . . . . . . . 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 422	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ ℝ) → (((1st ‘𝑦) = 𝑤 ∧ 𝑦 <ℝ 𝑥) → 𝑤 <R (1st ‘𝑥)))
20	19	ancomsd 466	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ ℝ) → ((𝑦 <ℝ 𝑥 ∧ (1st ‘𝑦) = 𝑤) → 𝑤 <R (1st ‘𝑥)))
21	20	an32s 650	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → ((𝑦 <ℝ 𝑥 ∧ (1st ‘𝑦) = 𝑤) → 𝑤 <R (1st ‘𝑥)))
22	21	rexlimdva 3155	. . . . . . . . . . . 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 416	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 → (∃𝑦 ∈ 𝐴 (1st ‘𝑦) = 𝑤 → 𝑤 <R (1st ‘𝑥))))
25	10, 24	syl7bi 254	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 → (𝑤 ∈ (1st “ 𝐴) → 𝑤 <R (1st ‘𝑥))))
26	25	impr 455	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) → (𝑤 ∈ (1st “ 𝐴) → 𝑤 <R (1st ‘𝑥)))
27	26	adantlr 713	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) → (𝑤 ∈ (1st “ 𝐴) → 𝑤 <R (1st ‘𝑥)))
28	27	ralrimiv 3145	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) → ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R (1st ‘𝑥))
29	28	expr 457	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 → ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R (1st ‘𝑥)))
30		brralrspcev 5208	. . . . 5 ⊢ (((1st ‘𝑥) ∈ R ∧ ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R (1st ‘𝑥)) → ∃𝑣 ∈ R ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R 𝑣)
31	3, 29, 30	syl6an 682	. . . 4 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 → ∃𝑣 ∈ R ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R 𝑣))
32	31	rexlimdva 3155	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 → ∃𝑣 ∈ R ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R 𝑣))
33		n0 4346	. . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐴)
34		fnfvima 7237	. . . . . . . . 9 ⊢ ((1st Fn V ∧ 𝐴 ⊆ V ∧ 𝑦 ∈ 𝐴) → (1st ‘𝑦) ∈ (1st “ 𝐴))
35	7, 8, 34	mp3an12 1451	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (1st ‘𝑦) ∈ (1st “ 𝐴))
36	35	ne0d 4335	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → (1st “ 𝐴) ≠ ∅)
37	36	exlimiv 1933	. . . . . 6 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (1st “ 𝐴) ≠ ∅)
38	33, 37	sylbi 216	. . . . 5 ⊢ (𝐴 ≠ ∅ → (1st “ 𝐴) ≠ ∅)
39		supsr 11109	. . . . . 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 413	. . . . 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 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))))
43		breq2 5152	. . . . . . . . . . . 12 ⊢ (𝑤 = (1st ‘𝑦) → (𝑣 <R 𝑤 ↔ 𝑣 <R (1st ‘𝑦)))
44	43	notbid 317	. . . . . . . . . . 11 ⊢ (𝑤 = (1st ‘𝑦) → (¬ 𝑣 <R 𝑤 ↔ ¬ 𝑣 <R (1st ‘𝑦)))
45	44	rspccv 3609	. . . . . . . . . 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 482	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 → ¬ 𝑣 <R (1st ‘𝑦)))
48		elreal2 11129	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ ↔ ((1st ‘𝑦) ∈ R ∧ 𝑦 = ⟨(1st ‘𝑦), 0R⟩))
49	48	simprbi 497	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 𝑦 = ⟨(1st ‘𝑦), 0R⟩)
50	49	breq2d 5160	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → (⟨𝑣, 0R⟩ <ℝ 𝑦 ↔ ⟨𝑣, 0R⟩ <ℝ ⟨(1st ‘𝑦), 0R⟩))
51		ltresr 11137	. . . . . . . . . . 11 ⊢ (⟨𝑣, 0R⟩ <ℝ ⟨(1st ‘𝑦), 0R⟩ ↔ 𝑣 <R (1st ‘𝑦))
52	50, 51	bitrdi 286	. . . . . . . . . 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 317	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → (¬ ⟨𝑣, 0R⟩ <ℝ 𝑦 ↔ ¬ 𝑣 <R (1st ‘𝑦)))
55	47, 54	sylibrd 258	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 → ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦))
56	55	ralrimdva 3154	. . . . . 6 ⊢ (𝐴 ⊆ ℝ → (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 → ∀𝑦 ∈ 𝐴 ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦))
57	56	ad2antrr 724	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣 ∈ R) → (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 → ∀𝑦 ∈ 𝐴 ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦))
58	49	breq1d 5158	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ → (𝑦 <ℝ ⟨𝑣, 0R⟩ ↔ ⟨(1st ‘𝑦), 0R⟩ <ℝ ⟨𝑣, 0R⟩))
59		ltresr 11137	. . . . . . . . . . . . . 14 ⊢ (⟨(1st ‘𝑦), 0R⟩ <ℝ ⟨𝑣, 0R⟩ ↔ (1st ‘𝑦) <R 𝑣)
60	58, 59	bitrdi 286	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → (𝑦 <ℝ ⟨𝑣, 0R⟩ ↔ (1st ‘𝑦) <R 𝑣))
61	48	simplbi 498	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ → (1st ‘𝑦) ∈ R)
62		breq1 5151	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (1st ‘𝑦) → (𝑤 <R 𝑣 ↔ (1st ‘𝑦) <R 𝑣))
63		breq1 5151	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (1st ‘𝑦) → (𝑤 <R 𝑢 ↔ (1st ‘𝑦) <R 𝑢))
64	63	rexbidv 3178	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (1st ‘𝑦) → (∃𝑢 ∈ (1st “ 𝐴)𝑤 <R 𝑢 ↔ ∃𝑢 ∈ (1st “ 𝐴)(1st ‘𝑦) <R 𝑢))
65	62, 64	imbi12d 344	. . . . . . . . . . . . . . . 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 3609	. . . . . . . . . . . . . . 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 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)))
71		fvelimab 6964	. . . . . . . . . . . . . . . 16 ⊢ ((1st Fn V ∧ 𝐴 ⊆ V) → (𝑢 ∈ (1st “ 𝐴) ↔ ∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑢))
72	7, 8, 71	mp2an 690	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (1st “ 𝐴) ↔ ∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑢)
73		ssel2 3977	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ)
74		ltresr2 11138	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑦 <ℝ 𝑧 ↔ (1st ‘𝑦) <R (1st ‘𝑧)))
75	73, 74	sylan2 593	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴)) → (𝑦 <ℝ 𝑧 ↔ (1st ‘𝑦) <R (1st ‘𝑧)))
76		breq2 5152	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1st ‘𝑧) = 𝑢 → ((1st ‘𝑦) <R (1st ‘𝑧) ↔ (1st ‘𝑦) <R 𝑢))
77	75, 76	sylan9bb 510	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴)) ∧ (1st ‘𝑧) = 𝑢) → (𝑦 <ℝ 𝑧 ↔ (1st ‘𝑦) <R 𝑢))
78	77	exbiri 809	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴)) → ((1st ‘𝑧) = 𝑢 → ((1st ‘𝑦) <R 𝑢 → 𝑦 <ℝ 𝑧)))
79	78	expr 457	. . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . 16 ⊢ (((1st ‘𝑦) <R 𝑢 ∧ (𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ)) → (𝑧 ∈ 𝐴 → ((1st ‘𝑧) = 𝑢 → 𝑦 <ℝ 𝑧)))
82	81	reximdvai 3165	. . . . . . . . . . . . . . 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 414	. . . . . . . . . . . . 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 3153	. . . . . . . . . . 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 413	. . . . . . . 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 724	. . . . . 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 3152	. . . . 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 11127	. . . . . . . 8 ⊢ (⟨𝑣, 0R⟩ ∈ ℝ ↔ 𝑣 ∈ R)
94	93	biimpri 227	. . . . . . 7 ⊢ (𝑣 ∈ R → ⟨𝑣, 0R⟩ ∈ ℝ)
95	94	adantl 482	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣 ∈ R) → ⟨𝑣, 0R⟩ ∈ ℝ)
96		breq1 5151	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝑣, 0R⟩ → (𝑥 <ℝ 𝑦 ↔ ⟨𝑣, 0R⟩ <ℝ 𝑦))
97	96	notbid 317	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑣, 0R⟩ → (¬ 𝑥 <ℝ 𝑦 ↔ ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦))
98	97	ralbidv 3177	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑣, 0R⟩ → (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦))
99		breq2 5152	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝑣, 0R⟩ → (𝑦 <ℝ 𝑥 ↔ 𝑦 <ℝ ⟨𝑣, 0R⟩))
100	99	imbi1d 341	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑣, 0R⟩ → ((𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧) ↔ (𝑦 <ℝ ⟨𝑣, 0R⟩ → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
101	100	ralbidv 3177	. . . . . . . . 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 631	. . . . . . . 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 3612	. . . . . . 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 413	. . . . . 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 608	. . . 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 3155	. . 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 1117	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 396   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  Vcvv 3474   ⊆ wss 3948  ∅c0 4322  ⟨cop 4634   class class class wbr 5148   “ cima 5679   Fn wfn 6538  ⟶wf 6539  –onto→wfo 6541  ‘cfv 6543  1^st c1st 7975  Rcnr 10862  0_Rc0r 10863   <_R cltr 10868  ℝcr 11111   <_ℝ cltrr 11116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-omul 8473  df-er 8705  df-ec 8707  df-qs 8711  df-ni 10869  df-pli 10870  df-mi 10871  df-lti 10872  df-plpq 10905  df-mpq 10906  df-ltpq 10907  df-enq 10908  df-nq 10909  df-erq 10910  df-plq 10911  df-mq 10912  df-1nq 10913  df-rq 10914  df-ltnq 10915  df-np 10978  df-1p 10979  df-plp 10980  df-mp 10981  df-ltp 10982  df-enr 11052  df-nr 11053  df-plr 11054  df-mr 11055  df-ltr 11056  df-0r 11057  df-1r 11058  df-m1r 11059  df-r 11122  df-lt 11125

This theorem is referenced by: (None)