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Theorem cf0 10187
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 10185 . . 3 (cf‘∅) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))}
2 0ss 4356 . . . . . . . . . . . 12 ∅ ⊆ 𝑦
32biantru 530 . . . . . . . . . . 11 (𝑦 ⊆ ∅ ↔ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))
4 ss0b 4357 . . . . . . . . . . 11 (𝑦 ⊆ ∅ ↔ 𝑦 = ∅)
53, 4bitr3i 276 . . . . . . . . . 10 ((𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦) ↔ 𝑦 = ∅)
65anbi1ci 626 . . . . . . . . 9 ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦)) ↔ (𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
76exbii 1850 . . . . . . . 8 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦)) ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
8 0ex 5264 . . . . . . . . 9 ∅ ∈ V
9 fveq2 6842 . . . . . . . . . 10 (𝑦 = ∅ → (card‘𝑦) = (card‘∅))
109eqeq2d 2747 . . . . . . . . 9 (𝑦 = ∅ → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘∅)))
118, 10ceqsexv 3494 . . . . . . . 8 (∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)) ↔ 𝑥 = (card‘∅))
12 card0 9894 . . . . . . . . 9 (card‘∅) = ∅
1312eqeq2i 2749 . . . . . . . 8 (𝑥 = (card‘∅) ↔ 𝑥 = ∅)
147, 11, 133bitri 296 . . . . . . 7 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦)) ↔ 𝑥 = ∅)
1514abbii 2806 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))} = {𝑥𝑥 = ∅}
16 df-sn 4587 . . . . . 6 {∅} = {𝑥𝑥 = ∅}
1715, 16eqtr4i 2767 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))} = {∅}
1817inteqi 4911 . . . 4 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))} = {∅}
198intsn 4947 . . . 4 {∅} = ∅
2018, 19eqtri 2764 . . 3 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))} = ∅
211, 20sseqtri 3980 . 2 (cf‘∅) ⊆ ∅
22 ss0b 4357 . 2 ((cf‘∅) ⊆ ∅ ↔ (cf‘∅) = ∅)
2321, 22mpbi 229 1 (cf‘∅) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wex 1781  {cab 2713  wss 3910  c0 4282  {csn 4586   cuni 4865   cint 4907  cfv 6496  cardccrd 9871  cfccf 9873
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-en 8884  df-card 9875  df-cf 9877
This theorem is referenced by:  cfeq0  10192  cflim2  10199  cfidm  10211  alephsing  10212  alephreg  10518  pwcfsdom  10519  rankcf  10713
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