Theorem cf0 9073

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 9071	. . 3 ⊢ (cf‘∅) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))}
2		0ss 3972	. . . . . . . . . . . . 13 ⊢ ∅ ⊆ ∪ 𝑦
3	2	biantru 526	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ ∅ ↔ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))
4		ss0b 3973	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ ∅ ↔ 𝑦 = ∅)
5	3, 4	bitr3i 266	. . . . . . . . . . 11 ⊢ ((𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦) ↔ 𝑦 = ∅)
6	5	anbi2i 730	. . . . . . . . . 10 ⊢ ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ (𝑥 = (card‘𝑦) ∧ 𝑦 = ∅))
7		ancom 466	. . . . . . . . . 10 ⊢ ((𝑥 = (card‘𝑦) ∧ 𝑦 = ∅) ↔ (𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
8	6, 7	bitri 264	. . . . . . . . 9 ⊢ ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ (𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
9	8	exbii 1774	. . . . . . . 8 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
10		0ex 4790	. . . . . . . . . 10 ⊢ ∅ ∈ V
11		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (card‘𝑦) = (card‘∅))
12	11	eqeq2d 2632	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘∅)))
13	10, 12	ceqsexv 3242	. . . . . . . . 9 ⊢ (∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)) ↔ 𝑥 = (card‘∅))
14		card0 8784	. . . . . . . . . 10 ⊢ (card‘∅) = ∅
15	14	eqeq2i 2634	. . . . . . . . 9 ⊢ (𝑥 = (card‘∅) ↔ 𝑥 = ∅)
16	13, 15	bitri 264	. . . . . . . 8 ⊢ (∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)) ↔ 𝑥 = ∅)
17	9, 16	bitri 264	. . . . . . 7 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ 𝑥 = ∅)
18	17	abbii 2739	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = {𝑥 ∣ 𝑥 = ∅}
19		df-sn 4178	. . . . . 6 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
20	18, 19	eqtr4i 2647	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = {∅}
21	20	inteqi 4479	. . . 4 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = ∩ {∅}
22	10	intsn 4513	. . . 4 ⊢ ∩ {∅} = ∅
23	21, 22	eqtri 2644	. . 3 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = ∅
24	1, 23	sseqtri 3637	. 2 ⊢ (cf‘∅) ⊆ ∅
25		ss0b 3973	. 2 ⊢ ((cf‘∅) ⊆ ∅ ↔ (cf‘∅) = ∅)
26	24, 25	mpbi 220	1 ⊢ (cf‘∅) = ∅

Syntax hints:   ∧ wa 384   = wceq 1483  ∃wex 1704  {cab 2608   ⊆ wss 3574  ∅c0 3915  {csn 4177  ∪ cuni 4436  ∩ cint 4475  ‘cfv 5888  cardccrd 8761  cfccf 8763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-en 7956  df-card 8765  df-cf 8767

This theorem is referenced by:  cfeq0  9078  cflim2  9085  cfidm  9097  alephsing  9098  alephreg  9404  pwcfsdom  9405  rankcf  9599