MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cf0 Structured version   Visualization version   GIF version

Theorem cf0 8956
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 8954 . . 3 (cf‘∅) ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))}
2 0ss 3924 . . . . . . . . . . . . 13 ∅ ⊆ 𝑦
32biantru 525 . . . . . . . . . . . 12 (𝑦 ⊆ ∅ ↔ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))
4 ss0b 3925 . . . . . . . . . . . 12 (𝑦 ⊆ ∅ ↔ 𝑦 = ∅)
53, 4bitr3i 265 . . . . . . . . . . 11 ((𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦) ↔ 𝑦 = ∅)
65anbi2i 726 . . . . . . . . . 10 ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦)) ↔ (𝑥 = (card‘𝑦) ∧ 𝑦 = ∅))
7 ancom 465 . . . . . . . . . 10 ((𝑥 = (card‘𝑦) ∧ 𝑦 = ∅) ↔ (𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
86, 7bitri 263 . . . . . . . . 9 ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦)) ↔ (𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
98exbii 1764 . . . . . . . 8 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦)) ↔ ∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)))
10 0ex 4718 . . . . . . . . . 10 ∅ ∈ V
11 fveq2 6103 . . . . . . . . . . 11 (𝑦 = ∅ → (card‘𝑦) = (card‘∅))
1211eqeq2d 2620 . . . . . . . . . 10 (𝑦 = ∅ → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘∅)))
1310, 12ceqsexv 3215 . . . . . . . . 9 (∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)) ↔ 𝑥 = (card‘∅))
14 card0 8667 . . . . . . . . . 10 (card‘∅) = ∅
1514eqeq2i 2622 . . . . . . . . 9 (𝑥 = (card‘∅) ↔ 𝑥 = ∅)
1613, 15bitri 263 . . . . . . . 8 (∃𝑦(𝑦 = ∅ ∧ 𝑥 = (card‘𝑦)) ↔ 𝑥 = ∅)
179, 16bitri 263 . . . . . . 7 (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦)) ↔ 𝑥 = ∅)
1817abbii 2726 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))} = {𝑥𝑥 = ∅}
19 df-sn 4126 . . . . . 6 {∅} = {𝑥𝑥 = ∅}
2018, 19eqtr4i 2635 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))} = {∅}
2120inteqi 4414 . . . 4 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))} = {∅}
2210intsn 4448 . . . 4 {∅} = ∅
2321, 22eqtri 2632 . . 3 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ ∅ ∧ ∅ ⊆ 𝑦))} = ∅
241, 23sseqtri 3600 . 2 (cf‘∅) ⊆ ∅
25 ss0b 3925 . 2 ((cf‘∅) ⊆ ∅ ↔ (cf‘∅) = ∅)
2624, 25mpbi 219 1 (cf‘∅) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1475  wex 1695  {cab 2596  wss 3540  c0 3874  {csn 4125   cuni 4372   cint 4410  cfv 5804  cardccrd 8644  cfccf 8646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-en 7842  df-card 8648  df-cf 8650
This theorem is referenced by:  cfeq0  8961  cflim2  8968  cfidm  8980  alephsing  8981  alephreg  9283  pwcfsdom  9284  rankcf  9478
  Copyright terms: Public domain W3C validator