Theorem coflim 10206

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 2821	. . . . 5 ⊢ (∪ 𝐵 = 𝐴 → (𝑥 ∈ ∪ 𝐵 ↔ 𝑥 ∈ 𝐴))
2	1	biimprd 247	. . . 4 ⊢ (∪ 𝐵 = 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐵))
3		eluni2 4874	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦)
4		limord 6382	. . . . . . . . 9 ⊢ (Lim 𝐴 → Ord 𝐴)
5		ssel2 3942	. . . . . . . . 9 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐴)
6		ordelon 6346	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
7	4, 5, 6	syl2an 596	. . . . . . . 8 ⊢ ((Lim 𝐴 ∧ (𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ On)
8	7	expr 457	. . . . . . 7 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑦 ∈ 𝐵 → 𝑦 ∈ On))
9		onelss 6364	. . . . . . 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 3158	. . . . 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 3145	. 2 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝐵 = 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
15		uniss 4878	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → ∪ 𝐵 ⊆ ∪ 𝐴)
16	15	3ad2ant2 1134	. . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) → ∪ 𝐵 ⊆ ∪ 𝐴)
17		uniss2 4907	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∪ 𝐴 ⊆ ∪ 𝐵)
18	17	3ad2ant3 1135	. . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) → ∪ 𝐴 ⊆ ∪ 𝐵)
19	16, 18	eqssd 3964	. . . 4 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) → ∪ 𝐵 = ∪ 𝐴)
20		limuni 6383	. . . . 5 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
21	20	3ad2ant1 1133	. . . 4 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) → 𝐴 = ∪ 𝐴)
22	19, 21	eqtr4d 2774	. . 3 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) → ∪ 𝐵 = 𝐴)
23	22	3expia 1121	. 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3060  ∃wrex 3069   ⊆ wss 3913  ∪ cuni 4870  Ord word 6321  Oncon0 6322  Lim wlim 6323

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-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326  df-lim 6327

This theorem is referenced by:  cflim3  10207  pwcfsdom  10528