Theorem coflim 9714

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 2841	. . . . 5 ⊢ (∪ 𝐵 = 𝐴 → (𝑥 ∈ ∪ 𝐵 ↔ 𝑥 ∈ 𝐴))
2	1	biimprd 251	. . . 4 ⊢ (∪ 𝐵 = 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐵))
3		eluni2 4803	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦)
4		limord 6229	. . . . . . . . 9 ⊢ (Lim 𝐴 → Ord 𝐴)
5		ssel2 3888	. . . . . . . . 9 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐴)
6		ordelon 6194	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
7	4, 5, 6	syl2an 599	. . . . . . . 8 ⊢ ((Lim 𝐴 ∧ (𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ On)
8	7	expr 461	. . . . . . 7 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑦 ∈ 𝐵 → 𝑦 ∈ On))
9		onelss 6212	. . . . . . 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 3197	. . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
12	3, 11	syl5bi 245	. . . 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 3118	. 2 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝐵 = 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
15		uniss 4807	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → ∪ 𝐵 ⊆ ∪ 𝐴)
16	15	3ad2ant2 1132	. . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) → ∪ 𝐵 ⊆ ∪ 𝐴)
17		uniss2 4834	. . . . . 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 3910	. . . 4 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) → ∪ 𝐵 = ∪ 𝐴)
20		limuni 6230	. . . . 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 2797	. . 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 215	1 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝐵 = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  ∀wral 3071  ∃wrex 3072   ⊆ wss 3859  ∪ cuni 4799  Ord word 6169  Oncon0 6170  Lim wlim 6171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-tr 5140  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-ord 6173  df-on 6174  df-lim 6175

This theorem is referenced by:  cflim3  9715  pwcfsdom  10036