Theorem cf0 10252

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 10250	. . 3 ⊢ (cf‘∅) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))}
2		0ss 4360	. . . . . . . . . . . 12 ⊢ ∅ ⊆ ∪ 𝑦
3	2	biantru 539	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ ∅ ↔ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))
4		ss0b 4361	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ ∅ ↔ 𝑦 = ∅)
5	3, 4	bitr3i 280	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦) ↔ 𝑦 = ∅)
6	5	anbi1ci 638	. . . . . . . . 9 ⊢ ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ (𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
7	6	exbii 1881	. . . . . . . 8 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
8		0ex 5275	. . . . . . . . 9 ⊢ ∅ ∈ V
9		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → (card‘𝑦) = (card‘∅))
10	9	eqeq2d 2777	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘∅)))
11	8, 10	ceqsexv 3506	. . . . . . . 8 ⊢ (∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)) ↔ 𝑥 = (card‘∅))
12		card0 9963	. . . . . . . . 9 ⊢ (card‘∅) = ∅
13	12	eqeq2i 2779	. . . . . . . 8 ⊢ (𝑥 = (card‘∅) ↔ 𝑥 = ∅)
14	7, 11, 13	3bitri 300	. . . . . . 7 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ 𝑥 = ∅)
15	14	abbii 2833	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = {𝑥 ∣ 𝑥 = ∅}
16		df-sn 4595	. . . . . 6 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
17	15, 16	eqtr4i 2792	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = {∅}
18	17	inteqi 4921	. . . 4 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = ∩ {∅}
19	8	intsn 4954	. . . 4 ⊢ ∩ {∅} = ∅
20	18, 19	eqtri 2789	. . 3 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = ∅
21	1, 20	sseqtri 3988	. 2 ⊢ (cf‘∅) ⊆ ∅
22		ss0b 4361	. 2 ⊢ ((cf‘∅) ⊆ ∅ ↔ (cf‘∅) = ∅)
23	21, 22	mpbi 233	1 ⊢ (cf‘∅) = ∅

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812  {cab 2744   ⊆ wss 3908  ∅c0 4289  {csn 4594  ∪ cuni 4877  ∩ cint 4917  ‘cfv 6543  cardccrd 9940  cfccf 9942

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ord 6370  df-on 6371  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-en 8953  df-card 9944  df-cf 9946

This theorem is used by:  cfeq0  10258  cflim2  10265  cfidm  10277  alephsing  10278  alephreg  10585  pwcfsdom  10586  rankcf  10780