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

Theorem cardom 8695
 Description: The set of natural numbers is a cardinal number. Theorem 18.11 of [Monk1] p. 133. (Contributed by NM, 28-Oct-2003.)
Assertion
Ref Expression
cardom (card‘ω) = ω

Proof of Theorem cardom
StepHypRef Expression
1 omelon 8426 . . . 4 ω ∈ On
2 oncardid 8665 . . . 4 (ω ∈ On → (card‘ω) ≈ ω)
31, 2ax-mp 5 . . 3 (card‘ω) ≈ ω
4 nnsdom 8434 . . . 4 ((card‘ω) ∈ ω → (card‘ω) ≺ ω)
5 sdomnen 7870 . . . 4 ((card‘ω) ≺ ω → ¬ (card‘ω) ≈ ω)
64, 5syl 17 . . 3 ((card‘ω) ∈ ω → ¬ (card‘ω) ≈ ω)
73, 6mt2 190 . 2 ¬ (card‘ω) ∈ ω
8 cardonle 8666 . . . 4 (ω ∈ On → (card‘ω) ⊆ ω)
91, 8ax-mp 5 . . 3 (card‘ω) ⊆ ω
10 cardon 8653 . . . 4 (card‘ω) ∈ On
1110, 1onsseli 5759 . . 3 ((card‘ω) ⊆ ω ↔ ((card‘ω) ∈ ω ∨ (card‘ω) = ω))
129, 11mpbi 219 . 2 ((card‘ω) ∈ ω ∨ (card‘ω) = ω)
137, 12mtpor 1686 1 (card‘ω) = ω
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 382   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540   class class class wbr 4583  Oncon0 5640  ‘cfv 5804  ωcom 6957   ≈ cen 7838   ≺ csdm 7840  cardccrd 8644 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  ax-inf2 8421 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-lim 5645  df-suc 5646  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-om 6958  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648 This theorem is referenced by:  infxpidm2  8723  alephcard  8776  infenaleph  8797  alephval2  9273  pwfseqlem5  9364
 Copyright terms: Public domain W3C validator