Theorem coflim 10219

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 2852	. . . . 5 ⊢ (∪ 𝐵 = 𝐴 → (𝑥 ∈ ∪ 𝐵 ↔ 𝑥 ∈ 𝐴))
2	1	biimprd 250	. . . 4 ⊢ (∪ 𝐵 = 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝐵))
3		eluni2 4870	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦)
4		limord 6408	. . . . . . . . 9 ⊢ (Lim 𝐴 → Ord 𝐴)
5		ssel2 3932	. . . . . . . . 9 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐴)
6		ordelon 6371	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
7	4, 5, 6	syl2an 605	. . . . . . . 8 ⊢ ((Lim 𝐴 ∧ (𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ On)
8	7	expr 460	. . . . . . 7 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑦 ∈ 𝐵 → 𝑦 ∈ On))
9		onelss 6389	. . . . . . 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 3174	. . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 → ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
12	3, 11	biimtrid 244	. . . 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 3161	. 2 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝐵 = 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))
15		uniss 4874	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → ∪ 𝐵 ⊆ ∪ 𝐴)
16	15	3ad2ant2 1148	. . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) → ∪ 𝐵 ⊆ ∪ 𝐴)
17		uniss2 4901	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∪ 𝐴 ⊆ ∪ 𝐵)
18	17	3ad2ant3 1149	. . . . 5 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) → ∪ 𝐴 ⊆ ∪ 𝐵)
19	16, 18	eqssd 3954	. . . 4 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) → ∪ 𝐵 = ∪ 𝐴)
20		limuni 6409	. . . . 5 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
21	20	3ad2ant1 1147	. . . 4 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) → 𝐴 = ∪ 𝐴)
22	19, 21	eqtr4d 2801	. . 3 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) → ∪ 𝐵 = 𝐴)
23	22	3expia 1135	. 2 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∪ 𝐵 = 𝐴))
24	14, 23	impbid 214	1 ⊢ ((Lim 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝐵 = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143  ∀wral 3077  ∃wrex 3087   ⊆ wss 3905  ∪ cuni 4866  Ord word 6346  Oncon0 6347  Lim wlim 6348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-tr 5209  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-ord 6350  df-on 6351  df-lim 6352

This theorem is referenced by:  cflim3  10220  pwcfsdom  10542