Theorem cf0 9471

Description: Value of the cofinality function at 0. Exercise 2 of [TakeutiZaring] p. 102. (Contributed by NM, 16-Apr-2004.)

Assertion

Ref	Expression
cf0	⊢ (cf‘∅) = ∅

Proof of Theorem cf0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cfub 9469	. . 3 ⊢ (cf‘∅) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))}
2		0ss 4236	. . . . . . . . . . . 12 ⊢ ∅ ⊆ ∪ 𝑦
3	2	biantru 522	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ ∅ ↔ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))
4		ss0b 4237	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ ∅ ↔ 𝑦 = ∅)
5	3, 4	bitr3i 269	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦) ↔ 𝑦 = ∅)
6	5	anbi1ci 616	. . . . . . . . 9 ⊢ ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ (𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
7	6	exbii 1810	. . . . . . . 8 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
8		0ex 5068	. . . . . . . . 9 ⊢ ∅ ∈ V
9		fveq2 6499	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → (card‘𝑦) = (card‘∅))
10	9	eqeq2d 2788	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘∅)))
11	8, 10	ceqsexv 3462	. . . . . . . 8 ⊢ (∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)) ↔ 𝑥 = (card‘∅))
12		card0 9181	. . . . . . . . 9 ⊢ (card‘∅) = ∅
13	12	eqeq2i 2790	. . . . . . . 8 ⊢ (𝑥 = (card‘∅) ↔ 𝑥 = ∅)
14	7, 11, 13	3bitri 289	. . . . . . 7 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ 𝑥 = ∅)
15	14	abbii 2844	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = {𝑥 ∣ 𝑥 = ∅}
16		df-sn 4442	. . . . . 6 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
17	15, 16	eqtr4i 2805	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = {∅}
18	17	inteqi 4753	. . . 4 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = ∩ {∅}
19	8	intsn 4785	. . . 4 ⊢ ∩ {∅} = ∅
20	18, 19	eqtri 2802	. . 3 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = ∅
21	1, 20	sseqtri 3893	. 2 ⊢ (cf‘∅) ⊆ ∅
22		ss0b 4237	. 2 ⊢ ((cf‘∅) ⊆ ∅ ↔ (cf‘∅) = ∅)
23	21, 22	mpbi 222	1 ⊢ (cf‘∅) = ∅

Syntax hints:   ∧ wa 387   = wceq 1507  ∃wex 1742  {cab 2758   ⊆ wss 3829  ∅c0 4178  {csn 4441  ∪ cuni 4712  ∩ cint 4749  ‘cfv 6188  cardccrd 9158  cfccf 9160

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-ord 6032  df-on 6033  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-en 8307  df-card 9162  df-cf 9164

This theorem is referenced by:  cfeq0  9476  cflim2  9483  cfidm  9495  alephsing  9496  alephreg  9802  pwcfsdom  9803  rankcf  9997