Theorem axpre-sup 11284

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 11415. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 11308. (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 11247	. . . . . . 7 ⊢ (𝑥 ∈ ℝ ↔ ((1st ‘𝑥) ∈ R ∧ 𝑥 = ⟨(1st ‘𝑥), 0R⟩))
2	1	simplbi 502	. . . . . 6 ⊢ (𝑥 ∈ ℝ → (1st ‘𝑥) ∈ R)
3	2	adantl 487	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ) → (1st ‘𝑥) ∈ R)
4		fo1st 8048	. . . . . . . . . . . 12 ⊢ 1st :V⟶ontoV
5		fof 6821	. . . . . . . . . . . 12 ⊢ (1st :V⟶ontoV → 1st :V⟶V)
6		ffn 6734	. . . . . . . . . . . 12 ⊢ (1st :V⟶V → 1st Fn V)
7	4, 5, 6	mp2b 10	. . . . . . . . . . 11 ⊢ 1st Fn V
8		ssv 3937	. . . . . . . . . . 11 ⊢ 𝐴 ⊆ V
9		fvelimab 6983	. . . . . . . . . . 11 ⊢ ((1st Fn V ∧ 𝐴 ⊆ V) → (𝑤 ∈ (1st “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 (1st ‘𝑦) = 𝑤))
10	7, 8, 9	mp2an 705	. . . . . . . . . 10 ⊢ (𝑤 ∈ (1st “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 (1st ‘𝑦) = 𝑤)
11		r19.29 3129	. . . . . . . . . . . 12 ⊢ ((∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∃𝑦 ∈ 𝐴 (1st ‘𝑦) = 𝑤) → ∃𝑦 ∈ 𝐴 (𝑦 <ℝ 𝑥 ∧ (1st ‘𝑦) = 𝑤))
12		ssel2 3908	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ)
13		ltresr2 11256	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 <ℝ 𝑥 ↔ (1st ‘𝑦) <R (1st ‘𝑥)))
14		breq1 5109	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑦) = 𝑤 → ((1st ‘𝑦) <R (1st ‘𝑥) ↔ 𝑤 <R (1st ‘𝑥)))
15	13, 14	sylan9bb 519	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ (1st ‘𝑦) = 𝑤) → (𝑦 <ℝ 𝑥 ↔ 𝑤 <R (1st ‘𝑥)))
16	15	biimpd 232	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) ∧ (1st ‘𝑦) = 𝑤) → (𝑦 <ℝ 𝑥 → 𝑤 <R (1st ‘𝑥)))
17	16	exp31 425	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ → (𝑥 ∈ ℝ → ((1st ‘𝑦) = 𝑤 → (𝑦 <ℝ 𝑥 → 𝑤 <R (1st ‘𝑥)))))
18	12, 17	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ ℝ → ((1st ‘𝑦) = 𝑤 → (𝑦 <ℝ 𝑥 → 𝑤 <R (1st ‘𝑥)))))
19	18	imp4b 427	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ ℝ) → (((1st ‘𝑦) = 𝑤 ∧ 𝑦 <ℝ 𝑥) → 𝑤 <R (1st ‘𝑥)))
20	19	ancomsd 471	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ ℝ) → ((𝑦 <ℝ 𝑥 ∧ (1st ‘𝑦) = 𝑤) → 𝑤 <R (1st ‘𝑥)))
21	20	an32s 665	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → ((𝑦 <ℝ 𝑥 ∧ (1st ‘𝑦) = 𝑤) → 𝑤 <R (1st ‘𝑥)))
22	21	rexlimdva 3167	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∃𝑦 ∈ 𝐴 (𝑦 <ℝ 𝑥 ∧ (1st ‘𝑦) = 𝑤) → 𝑤 <R (1st ‘𝑥)))
23	11, 22	syl5 35	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → ((∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 ∧ ∃𝑦 ∈ 𝐴 (1st ‘𝑦) = 𝑤) → 𝑤 <R (1st ‘𝑥)))
24	23	expd 421	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 → (∃𝑦 ∈ 𝐴 (1st ‘𝑦) = 𝑤 → 𝑤 <R (1st ‘𝑥))))
25	10, 24	syl7bi 258	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 → (𝑤 ∈ (1st “ 𝐴) → 𝑤 <R (1st ‘𝑥))))
26	25	impr 460	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) → (𝑤 ∈ (1st “ 𝐴) → 𝑤 <R (1st ‘𝑥)))
27	26	adantlr 728	. . . . . . 7 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) → (𝑤 ∈ (1st “ 𝐴) → 𝑤 <R (1st ‘𝑥)))
28	27	ralrimiv 3157	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥)) → ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R (1st ‘𝑥))
29	28	expr 462	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 → ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R (1st ‘𝑥)))
30		brralrspcev 5169	. . . . 5 ⊢ (((1st ‘𝑥) ∈ R ∧ ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R (1st ‘𝑥)) → ∃𝑣 ∈ R ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R 𝑣)
31	3, 29, 30	syl6an 697	. . . 4 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 → ∃𝑣 ∈ R ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R 𝑣))
32	31	rexlimdva 3167	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥 → ∃𝑣 ∈ R ∀𝑤 ∈ (1st “ 𝐴)𝑤 <R 𝑣))
33		n0 4303	. . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐴)
34		fnfvima 7265	. . . . . . . . 9 ⊢ ((1st Fn V ∧ 𝐴 ⊆ V ∧ 𝑦 ∈ 𝐴) → (1st ‘𝑦) ∈ (1st “ 𝐴))
35	7, 8, 34	mp3an12 1481	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (1st ‘𝑦) ∈ (1st “ 𝐴))
36	35	ne0d 4291	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → (1st “ 𝐴) ≠ ∅)
37	36	exlimiv 1965	. . . . . 6 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (1st “ 𝐴) ≠ ∅)
38	33, 37	sylbi 220	. . . . 5 ⊢ (𝐴 ≠ ∅ → (1st “ 𝐴) ≠ ∅)
39		supsr 11227	. . . . . 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 418	. . . . 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 18	. . . 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 487	. . 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 5110	. . . . . . . . . . . 12 ⊢ (𝑤 = (1st ‘𝑦) → (𝑣 <R 𝑤 ↔ 𝑣 <R (1st ‘𝑦)))
44	43	notbid 321	. . . . . . . . . . 11 ⊢ (𝑤 = (1st ‘𝑦) → (¬ 𝑣 <R 𝑤 ↔ ¬ 𝑣 <R (1st ‘𝑦)))
45	44	rspccv 3577	. . . . . . . . . 10 ⊢ (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 → ((1st ‘𝑦) ∈ (1st “ 𝐴) → ¬ 𝑣 <R (1st ‘𝑦)))
46	35, 45	syl5com 32	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 → ¬ 𝑣 <R (1st ‘𝑦)))
47	46	adantl 487	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 → ¬ 𝑣 <R (1st ‘𝑦)))
48		elreal2 11247	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ ↔ ((1st ‘𝑦) ∈ R ∧ 𝑦 = ⟨(1st ‘𝑦), 0R⟩))
49	48	simprbi 503	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 𝑦 = ⟨(1st ‘𝑦), 0R⟩)
50	49	breq2d 5118	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → (⟨𝑣, 0R⟩ <ℝ 𝑦 ↔ ⟨𝑣, 0R⟩ <ℝ ⟨(1st ‘𝑦), 0R⟩))
51		ltresr 11255	. . . . . . . . . . 11 ⊢ (⟨𝑣, 0R⟩ <ℝ ⟨(1st ‘𝑦), 0R⟩ ↔ 𝑣 <R (1st ‘𝑦))
52	50, 51	bitrdi 290	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → (⟨𝑣, 0R⟩ <ℝ 𝑦 ↔ 𝑣 <R (1st ‘𝑦)))
53	12, 52	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → (⟨𝑣, 0R⟩ <ℝ 𝑦 ↔ 𝑣 <R (1st ‘𝑦)))
54	53	notbid 321	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → (¬ ⟨𝑣, 0R⟩ <ℝ 𝑦 ↔ ¬ 𝑣 <R (1st ‘𝑦)))
55	47, 54	sylibrd 262	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 → ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦))
56	55	ralrimdva 3166	. . . . . 6 ⊢ (𝐴 ⊆ ℝ → (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 → ∀𝑦 ∈ 𝐴 ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦))
57	56	ad2antrr 739	. . . . 5 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣 ∈ R) → (∀𝑤 ∈ (1st “ 𝐴) ¬ 𝑣 <R 𝑤 → ∀𝑦 ∈ 𝐴 ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦))
58	49	breq1d 5116	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ → (𝑦 <ℝ ⟨𝑣, 0R⟩ ↔ ⟨(1st ‘𝑦), 0R⟩ <ℝ ⟨𝑣, 0R⟩))
59		ltresr 11255	. . . . . . . . . . . . . 14 ⊢ (⟨(1st ‘𝑦), 0R⟩ <ℝ ⟨𝑣, 0R⟩ ↔ (1st ‘𝑦) <R 𝑣)
60	58, 59	bitrdi 290	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ → (𝑦 <ℝ ⟨𝑣, 0R⟩ ↔ (1st ‘𝑦) <R 𝑣))
61	48	simplbi 502	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℝ → (1st ‘𝑦) ∈ R)
62		breq1 5109	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (1st ‘𝑦) → (𝑤 <R 𝑣 ↔ (1st ‘𝑦) <R 𝑣))
63		breq1 5109	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (1st ‘𝑦) → (𝑤 <R 𝑢 ↔ (1st ‘𝑦) <R 𝑢))
64	63	rexbidv 3190	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (1st ‘𝑦) → (∃𝑢 ∈ (1st “ 𝐴)𝑤 <R 𝑢 ↔ ∃𝑢 ∈ (1st “ 𝐴)(1st ‘𝑦) <R 𝑢))
65	62, 64	imbi12d 347	. . . . . . . . . . . . . . . 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 3577	. . . . . . . . . . . . . . 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 35	. . . . . . . . . . . . . 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 90	. . . . . . . . . . . . 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 243	. . . . . . . . . . . 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 486	. . . . . . . . . . 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 6983	. . . . . . . . . . . . . . . 16 ⊢ ((1st Fn V ∧ 𝐴 ⊆ V) → (𝑢 ∈ (1st “ 𝐴) ↔ ∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑢))
72	7, 8, 71	mp2an 705	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (1st “ 𝐴) ↔ ∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑢)
73		ssel2 3908	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ)
74		ltresr2 11256	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑦 <ℝ 𝑧 ↔ (1st ‘𝑦) <R (1st ‘𝑧)))
75	73, 74	sylan2 605	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴)) → (𝑦 <ℝ 𝑧 ↔ (1st ‘𝑦) <R (1st ‘𝑧)))
76		breq2 5110	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1st ‘𝑧) = 𝑢 → ((1st ‘𝑦) <R (1st ‘𝑧) ↔ (1st ‘𝑦) <R 𝑢))
77	75, 76	sylan9bb 519	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴)) ∧ (1st ‘𝑧) = 𝑢) → (𝑦 <ℝ 𝑧 ↔ (1st ‘𝑦) <R 𝑢))
78	77	exbiri 823	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℝ ∧ (𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴)) → ((1st ‘𝑧) = 𝑢 → ((1st ‘𝑦) <R 𝑢 → 𝑦 <ℝ 𝑧)))
79	78	expr 462	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (𝑧 ∈ 𝐴 → ((1st ‘𝑧) = 𝑢 → ((1st ‘𝑦) <R 𝑢 → 𝑦 <ℝ 𝑧))))
80	79	com4r 95	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘𝑦) <R 𝑢 → ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (𝑧 ∈ 𝐴 → ((1st ‘𝑧) = 𝑢 → 𝑦 <ℝ 𝑧))))
81	80	imp 412	. . . . . . . . . . . . . . . 16 ⊢ (((1st ‘𝑦) <R 𝑢 ∧ (𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ)) → (𝑧 ∈ 𝐴 → ((1st ‘𝑧) = 𝑢 → 𝑦 <ℝ 𝑧)))
82	81	reximdvai 3177	. . . . . . . . . . . . . . 15 ⊢ (((1st ‘𝑦) <R 𝑢 ∧ (𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ)) → (∃𝑧 ∈ 𝐴 (1st ‘𝑧) = 𝑢 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))
83	72, 82	biimtrid 245	. . . . . . . . . . . . . 14 ⊢ (((1st ‘𝑦) <R 𝑢 ∧ (𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ)) → (𝑢 ∈ (1st “ 𝐴) → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))
84	83	expcom 419	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → ((1st ‘𝑦) <R 𝑢 → (𝑢 ∈ (1st “ 𝐴) → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
85	84	com23 87	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (𝑢 ∈ (1st “ 𝐴) → ((1st ‘𝑦) <R 𝑢 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
86	85	rexlimdv 3165	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ ∧ 𝐴 ⊆ ℝ) → (∃𝑢 ∈ (1st “ 𝐴)(1st ‘𝑦) <R 𝑢 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))
87	70, 86	syl6d 76	. . . . . . . . . 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 87	. . . . . . . . 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 418	. . . . . . . 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 90	. . . . . . 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 739	. . . . . 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 3164	. . . . 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 11245	. . . . . . 7 ⊢ (⟨𝑣, 0R⟩ ∈ ℝ ↔ 𝑣 ∈ R)
94	93	bilanri 512	. . . . . 6 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑣 ∈ R) → ⟨𝑣, 0R⟩ ∈ ℝ)
95		breq1 5109	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝑣, 0R⟩ → (𝑥 <ℝ 𝑦 ↔ ⟨𝑣, 0R⟩ <ℝ 𝑦))
96	95	notbid 321	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑣, 0R⟩ → (¬ 𝑥 <ℝ 𝑦 ↔ ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦))
97	96	ralbidv 3189	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑣, 0R⟩ → (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ ⟨𝑣, 0R⟩ <ℝ 𝑦))
98		breq2 5110	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝑣, 0R⟩ → (𝑦 <ℝ 𝑥 ↔ 𝑦 <ℝ ⟨𝑣, 0R⟩))
99	98	imbi1d 344	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑣, 0R⟩ → ((𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧) ↔ (𝑦 <ℝ ⟨𝑣, 0R⟩ → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧)))
100	99	ralbidv 3189	. . . . . . . . 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)))
101	97, 100	anbi12d 644	. . . . . . . 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))))
102	101	rspcev 3580	. . . . . . 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)))
103	102	ex 418	. . . . . 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))))
104	94, 103	syl 18	. . . . 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))))
105	57, 92, 104	syl2and 620	. . . 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))))
106	105	rexlimdva 3167	. . 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))))
107	32, 42, 106	3syld 61	. 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))))
108	107	3impia 1135	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)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  ⟨cop 4838   class class class wbr 5106   “ cima 5659   Fn wfn 6556  ⟶wf 6557  ⟶ontowfo 6559  ‘cfv 6561  1^st c1st 8026  Rcnr 10980  0_Rc0r 10981   <_R cltr 10986  ℝcr 11229   <_ℝ cltrr 11234

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-inf2 9672

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-oadd 8511  df-omul 8512  df-er 8748  df-ec 8750  df-qs 8754  df-ni 10987  df-pli 10988  df-mi 10989  df-lti 10990  df-plpq 11023  df-mpq 11024  df-ltpq 11025  df-enq 11026  df-nq 11027  df-erq 11028  df-plq 11029  df-mq 11030  df-1nq 11031  df-rq 11032  df-ltnq 11033  df-np 11096  df-1p 11097  df-plp 11098  df-mp 11099  df-ltp 11100  df-enr 11170  df-nr 11171  df-plr 11172  df-mr 11173  df-ltr 11174  df-0r 11175  df-1r 11176  df-m1r 11177  df-r 11240  df-lt 11243

This theorem is used by: (None)