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Theorem cf0 10292
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.
StepHypRef Expression
1 cfub 10290 . . 3 (cf‘∅) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))}
2 0ss 4399 . . . . . . . . . . . 12 ∅ ⊆ 𝑦
32biantru 529 . . . . . . . . . . 11 (𝑦 ⊆ ∅ ↔ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))
4 ss0b 4400 . . . . . . . . . . 11 (𝑦 ⊆ ∅ ↔ 𝑦 = ∅)
53, 4bitr3i 277 . . . . . . . . . 10 ((𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦) ↔ 𝑦 = ∅)
65anbi1ci 626 . . . . . . . . 9 ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦)) ↔ (𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
76exbii 1847 . . . . . . . 8 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦)) ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
8 0ex 5306 . . . . . . . . 9 ∅ ∈ V
9 fveq2 6905 . . . . . . . . . 10 (𝑦 = ∅ → (card‘𝑦) = (card‘∅))
109eqeq2d 2747 . . . . . . . . 9 (𝑦 = ∅ → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘∅)))
118, 10ceqsexv 3531 . . . . . . . 8 (∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)) ↔ 𝑥 = (card‘∅))
12 card0 9999 . . . . . . . . 9 (card‘∅) = ∅
1312eqeq2i 2749 . . . . . . . 8 (𝑥 = (card‘∅) ↔ 𝑥 = ∅)
147, 11, 133bitri 297 . . . . . . 7 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦)) ↔ 𝑥 = ∅)
1514abbii 2808 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))} = {𝑥𝑥 = ∅}
16 df-sn 4626 . . . . . 6 {∅} = {𝑥𝑥 = ∅}
1715, 16eqtr4i 2767 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))} = {∅}
1817inteqi 4949 . . . 4 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))} = {∅}
198intsn 4983 . . . 4 {∅} = ∅
2018, 19eqtri 2764 . . 3 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))} = ∅
211, 20sseqtri 4031 . 2 (cf‘∅) ⊆ ∅
22 ss0b 4400 . 2 ((cf‘∅) ⊆ ∅ ↔ (cf‘∅) = ∅)
2321, 22mpbi 230 1 (cf‘∅) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1539  wex 1778  {cab 2713  wss 3950  c0 4332  {csn 4625   cuni 4906   cint 4945  cfv 6560  cardccrd 9976  cfccf 9978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-en 8987  df-card 9980  df-cf 9982
This theorem is referenced by:  cfeq0  10297  cflim2  10304  cfidm  10316  alephsing  10317  alephreg  10623  pwcfsdom  10624  rankcf  10818
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