Theorem coflim 10260

Description: A simpler expression for the cofinality predicate, at a limit ordinal. (Contributed by Mario Carneiro, 28-Feb-2013.)

Assertion

Ref	Expression
coflim	⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝐵 = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem coflim

Step	Hyp	Ref	Expression
1		eleq2 2820	. . . . 5 ⊢ (∪ 𝐵 = 𝐴 → (𝑥 ∈ ∪ 𝐵 ↔ 𝑥 ∈ 𝐴))
2	1	biimprd 247	. . . 4 ⊢ (∪ 𝐵 = 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐵))
3		eluni2 4913	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦)
4		limord 6425	. . . . . . . . 9 ⊢ (Lim 𝐴 → Ord 𝐴)
5		ssel2 3978	. . . . . . . . 9 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐴)
6		ordelon 6389	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
7	4, 5, 6	syl2an 594	. . . . . . . 8 ⊢ ((Lim 𝐴 ∧ (𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ On)
8	7	expr 455	. . . . . . 7 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑦 ∈ 𝐵 → 𝑦 ∈ On))
9		onelss 6407	. . . . . . 7 ⊢ (𝑦 ∈ On → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦))
10	8, 9	syl6 35	. . . . . 6 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑦 ∈ 𝐵 → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦)))
11	10	reximdvai 3163	. . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
12	3, 11	biimtrid 241	. . . 4 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ ∪ 𝐵 → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
13	2, 12	syl9r 78	. . 3 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝐵 = 𝐴 → (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
14	13	ralrimdv 3150	. 2 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝐵 = 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
15		uniss 4917	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → ∪ 𝐵 ⊆ ∪ 𝐴)
16	15	3ad2ant2 1132	. . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) → ∪ 𝐵 ⊆ ∪ 𝐴)
17		uniss2 4946	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∪ 𝐴 ⊆ ∪ 𝐵)
18	17	3ad2ant3 1133	. . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) → ∪ 𝐴 ⊆ ∪ 𝐵)
19	16, 18	eqssd 4000	. . . 4 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) → ∪ 𝐵 = ∪ 𝐴)
20		limuni 6426	. . . . 5 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
21	20	3ad2ant1 1131	. . . 4 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) → 𝐴 = ∪ 𝐴)
22	19, 21	eqtr4d 2773	. . 3 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) → ∪ 𝐵 = 𝐴)
23	22	3expia 1119	. 2 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∪ 𝐵 = 𝐴))
24	14, 23	impbid 211	1 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝐵 = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  ∀wral 3059  ∃wrex 3068   ⊆ wss 3949  ∪ cuni 4909  Ord word 6364  Oncon0 6365  Lim wlim 6366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369  df-lim 6370

This theorem is referenced by:  cflim3  10261  pwcfsdom  10582