Theorem coflim 10266

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 2849	. . . . 5 ⊢ (∪ 𝐵 = 𝐴 → (𝑥 ∈ ∪ 𝐵 ↔ 𝑥 ∈ 𝐴))
2	1	biimprd 251	. . . 4 ⊢ (∪ 𝐵 = 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐵))
3		eluni2 4871	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦)
4		limord 6419	. . . . . . . . 9 ⊢ (Lim 𝐴 → Ord 𝐴)
5		ssel2 3926	. . . . . . . . 9 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐴)
6		ordelon 6381	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
7	4, 5, 6	syl2an 608	. . . . . . . 8 ⊢ ((Lim 𝐴 ∧ (𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ On)
8	7	expr 462	. . . . . . 7 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑦 ∈ 𝐵 → 𝑦 ∈ On))
9		onelss 6400	. . . . . . 7 ⊢ (𝑦 ∈ On → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦))
10	8, 9	syl6 36	. . . . . 6 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑦 ∈ 𝐵 → (𝑥 ∈ 𝑦 → 𝑥 ⊆ 𝑦)))
11	10	reximdvai 3173	. . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
12	3, 11	biimtrid 245	. . . 4 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ ∪ 𝐵 → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
13	2, 12	syl9r 79	. . 3 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝐵 = 𝐴 → (𝑥 ∈ 𝐴 → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)))
14	13	ralrimdv 3160	. 2 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝐵 = 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
15		uniss 4875	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → ∪ 𝐵 ⊆ ∪ 𝐴)
16	15	3ad2ant2 1152	. . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) → ∪ 𝐵 ⊆ ∪ 𝐴)
17		uniss2 4902	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∪ 𝐴 ⊆ ∪ 𝐵)
18	17	3ad2ant3 1153	. . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) → ∪ 𝐴 ⊆ ∪ 𝐵)
19	16, 18	eqssd 3948	. . . 4 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) → ∪ 𝐵 = ∪ 𝐴)
20		limuni 6420	. . . . 5 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
21	20	3ad2ant1 1151	. . . 4 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) → 𝐴 = ∪ 𝐴)
22	19, 21	eqtr4d 2798	. . 3 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) → ∪ 𝐵 = 𝐴)
23	22	3expia 1139	. 2 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∪ 𝐵 = 𝐴))
24	14, 23	impbid 215	1 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝐵 = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   ⊆ wss 3899  ∪ cuni 4867  Ord word 6356  Oncon0 6357  Lim wlim 6358

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6360  df-on 6361  df-lim 6362

This theorem is used by:  cflim3  10267  pwcfsdom  10595