Theorem cf0 10289

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 10287	. . 3 ⊢ (cf‘∅) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))}
2		0ss 4406	. . . . . . . . . . . 12 ⊢ ∅ ⊆ ∪ 𝑦
3	2	biantru 529	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ ∅ ↔ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))
4		ss0b 4407	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ ∅ ↔ 𝑦 = ∅)
5	3, 4	bitr3i 277	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦) ↔ 𝑦 = ∅)
6	5	anbi1ci 626	. . . . . . . . 9 ⊢ ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ (𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
7	6	exbii 1845	. . . . . . . 8 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
8		0ex 5313	. . . . . . . . 9 ⊢ ∅ ∈ V
9		fveq2 6907	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → (card‘𝑦) = (card‘∅))
10	9	eqeq2d 2746	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘∅)))
11	8, 10	ceqsexv 3530	. . . . . . . 8 ⊢ (∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)) ↔ 𝑥 = (card‘∅))
12		card0 9996	. . . . . . . . 9 ⊢ (card‘∅) = ∅
13	12	eqeq2i 2748	. . . . . . . 8 ⊢ (𝑥 = (card‘∅) ↔ 𝑥 = ∅)
14	7, 11, 13	3bitri 297	. . . . . . 7 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦)) ↔ 𝑥 = ∅)
15	14	abbii 2807	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = {𝑥 ∣ 𝑥 = ∅}
16		df-sn 4632	. . . . . 6 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
17	15, 16	eqtr4i 2766	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = {∅}
18	17	inteqi 4955	. . . 4 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = ∩ {∅}
19	8	intsn 4989	. . . 4 ⊢ ∩ {∅} = ∅
20	18, 19	eqtri 2763	. . 3 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ ∪ 𝑦))} = ∅
21	1, 20	sseqtri 4032	. 2 ⊢ (cf‘∅) ⊆ ∅
22		ss0b 4407	. 2 ⊢ ((cf‘∅) ⊆ ∅ ↔ (cf‘∅) = ∅)
23	21, 22	mpbi 230	1 ⊢ (cf‘∅) = ∅

Syntax hints:   ∧ wa 395   = wceq 1537  ∃wex 1776  {cab 2712   ⊆ wss 3963  ∅c0 4339  {csn 4631  ∪ cuni 4912  ∩ cint 4951  ‘cfv 6563  cardccrd 9973  cfccf 9975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-en 8985  df-card 9977  df-cf 9979

This theorem is referenced by:  cfeq0  10294  cflim2  10301  cfidm  10313  alephsing  10314  alephreg  10620  pwcfsdom  10621  rankcf  10815