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Theorem cf0 9666
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 9664 . . 3 (cf‘∅) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))}
2 0ss 4307 . . . . . . . . . . . 12 ∅ ⊆ 𝑦
32biantru 533 . . . . . . . . . . 11 (𝑦 ⊆ ∅ ↔ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))
4 ss0b 4308 . . . . . . . . . . 11 (𝑦 ⊆ ∅ ↔ 𝑦 = ∅)
53, 4bitr3i 280 . . . . . . . . . 10 ((𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦) ↔ 𝑦 = ∅)
65anbi1ci 628 . . . . . . . . 9 ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦)) ↔ (𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
76exbii 1849 . . . . . . . 8 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦)) ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
8 0ex 5178 . . . . . . . . 9 ∅ ∈ V
9 fveq2 6649 . . . . . . . . . 10 (𝑦 = ∅ → (card‘𝑦) = (card‘∅))
109eqeq2d 2812 . . . . . . . . 9 (𝑦 = ∅ → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘∅)))
118, 10ceqsexv 3492 . . . . . . . 8 (∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)) ↔ 𝑥 = (card‘∅))
12 card0 9375 . . . . . . . . 9 (card‘∅) = ∅
1312eqeq2i 2814 . . . . . . . 8 (𝑥 = (card‘∅) ↔ 𝑥 = ∅)
147, 11, 133bitri 300 . . . . . . 7 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦)) ↔ 𝑥 = ∅)
1514abbii 2866 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))} = {𝑥𝑥 = ∅}
16 df-sn 4529 . . . . . 6 {∅} = {𝑥𝑥 = ∅}
1715, 16eqtr4i 2827 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))} = {∅}
1817inteqi 4845 . . . 4 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))} = {∅}
198intsn 4877 . . . 4 {∅} = ∅
2018, 19eqtri 2824 . . 3 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))} = ∅
211, 20sseqtri 3954 . 2 (cf‘∅) ⊆ ∅
22 ss0b 4308 . 2 ((cf‘∅) ⊆ ∅ ↔ (cf‘∅) = ∅)
2321, 22mpbi 233 1 (cf‘∅) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wex 1781  {cab 2779  wss 3884  c0 4246  {csn 4528   cuni 4803   cint 4841  cfv 6328  cardccrd 9352  cfccf 9354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-en 8497  df-card 9356  df-cf 9358
This theorem is referenced by:  cfeq0  9671  cflim2  9678  cfidm  9690  alephsing  9691  alephreg  9997  pwcfsdom  9998  rankcf  10192
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