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Theorem cf0 8933
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 8931 . . 3 (cf‘∅) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))}
2 0ss 3923 . . . . . . . . . . . . 13 ∅ ⊆ 𝑦
32biantru 524 . . . . . . . . . . . 12 (𝑦 ⊆ ∅ ↔ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))
4 ss0b 3924 . . . . . . . . . . . 12 (𝑦 ⊆ ∅ ↔ 𝑦 = ∅)
53, 4bitr3i 264 . . . . . . . . . . 11 ((𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦) ↔ 𝑦 = ∅)
65anbi2i 725 . . . . . . . . . 10 ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦)) ↔ (𝑥 = (card‘𝑦) ∧ 𝑦 = ∅))
7 ancom 464 . . . . . . . . . 10 ((𝑥 = (card‘𝑦) ∧ 𝑦 = ∅) ↔ (𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
86, 7bitri 262 . . . . . . . . 9 ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦)) ↔ (𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
98exbii 1763 . . . . . . . 8 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦)) ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
10 0ex 4713 . . . . . . . . . 10 ∅ ∈ V
11 fveq2 6088 . . . . . . . . . . 11 (𝑦 = ∅ → (card‘𝑦) = (card‘∅))
1211eqeq2d 2619 . . . . . . . . . 10 (𝑦 = ∅ → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘∅)))
1310, 12ceqsexv 3214 . . . . . . . . 9 (∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)) ↔ 𝑥 = (card‘∅))
14 card0 8644 . . . . . . . . . 10 (card‘∅) = ∅
1514eqeq2i 2621 . . . . . . . . 9 (𝑥 = (card‘∅) ↔ 𝑥 = ∅)
1613, 15bitri 262 . . . . . . . 8 (∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)) ↔ 𝑥 = ∅)
179, 16bitri 262 . . . . . . 7 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦)) ↔ 𝑥 = ∅)
1817abbii 2725 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))} = {𝑥𝑥 = ∅}
19 df-sn 4125 . . . . . 6 {∅} = {𝑥𝑥 = ∅}
2018, 19eqtr4i 2634 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))} = {∅}
2120inteqi 4408 . . . 4 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))} = {∅}
2210intsn 4442 . . . 4 {∅} = ∅
2321, 22eqtri 2631 . . 3 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))} = ∅
241, 23sseqtri 3599 . 2 (cf‘∅) ⊆ ∅
25 ss0b 3924 . 2 ((cf‘∅) ⊆ ∅ ↔ (cf‘∅) = ∅)
2624, 25mpbi 218 1 (cf‘∅) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1474  wex 1694  {cab 2595  wss 3539  c0 3873  {csn 4124   cuni 4366   cint 4404  cfv 5790  cardccrd 8621  cfccf 8623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-en 7819  df-card 8625  df-cf 8627
This theorem is referenced by:  cfeq0  8938  cflim2  8945  cfidm  8957  alephsing  8958  alephreg  9260  pwcfsdom  9261  rankcf  9455
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