Theorem cf0 8933

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 8931	. . 3 ⊢ (cf‘∅) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))}
2		0ss 3923	. . . . . . . . . . . . 13 ⊢ ∅ ⊆ ∪ 𝑦
3	2	biantru 524	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ ∅ ↔ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))
4		ss0b 3924	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ ∅ ↔ 𝑦 = ∅)
5	3, 4	bitr3i 264	. . . . . . . . . . 11 ⊢ ((𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦) ↔ 𝑦 = ∅)
6	5	anbi2i 725	. . . . . . . . . 10 ⊢ ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ (𝑥 = (card‘𝑦) ∧ 𝑦 = ∅))
7		ancom 464	. . . . . . . . . 10 ⊢ ((𝑥 = (card‘𝑦) ∧ 𝑦 = ∅) ↔ (𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
8	6, 7	bitri 262	. . . . . . . . 9 ⊢ ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ (𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
9	8	exbii 1763	. . . . . . . 8 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
10		0ex 4713	. . . . . . . . . 10 ⊢ ∅ ∈ V
11		fveq2 6088	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (card‘𝑦) = (card‘∅))
12	11	eqeq2d 2619	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘∅)))
13	10, 12	ceqsexv 3214	. . . . . . . . 9 ⊢ (∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)) ↔ 𝑥 = (card‘∅))
14		card0 8644	. . . . . . . . . 10 ⊢ (card‘∅) = ∅
15	14	eqeq2i 2621	. . . . . . . . 9 ⊢ (𝑥 = (card‘∅) ↔ 𝑥 = ∅)
16	13, 15	bitri 262	. . . . . . . 8 ⊢ (∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)) ↔ 𝑥 = ∅)
17	9, 16	bitri 262	. . . . . . 7 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ 𝑥 = ∅)
18	17	abbii 2725	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = {𝑥 ∣ 𝑥 = ∅}
19		df-sn 4125	. . . . . 6 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
20	18, 19	eqtr4i 2634	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = {∅}
21	20	inteqi 4408	. . . 4 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = ∩ {∅}
22	10	intsn 4442	. . . 4 ⊢ ∩ {∅} = ∅
23	21, 22	eqtri 2631	. . 3 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = ∅
24	1, 23	sseqtri 3599	. 2 ⊢ (cf‘∅) ⊆ ∅
25		ss0b 3924	. 2 ⊢ ((cf‘∅) ⊆ ∅ ↔ (cf‘∅) = ∅)
26	24, 25	mpbi 218	1 ⊢ (cf‘∅) = ∅

Syntax hints:   ∧ wa 382   = wceq 1474  ∃wex 1694  {cab 2595   ⊆ wss 3539  ∅c0 3873  {csn 4124  ∪ cuni 4366  ∩ cint 4404  ‘cfv 5790  cardccrd 8621  cfccf 8623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-en 7819  df-card 8625  df-cf 8627

This theorem is referenced by:  cfeq0  8938  cflim2  8945  cfidm  8957  alephsing  8958  alephreg  9260  pwcfsdom  9261  rankcf  9455