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

Theorem cardom 9391
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 9085 . . . 4 ω ∈ On
2 oncardid 9361 . . . 4 (ω ∈ On → (card‘ω) ≈ ω)
31, 2ax-mp 5 . . 3 (card‘ω) ≈ ω
4 nnsdom 9093 . . . 4 ((card‘ω) ∈ ω → (card‘ω) ≺ ω)
5 sdomnen 8513 . . . 4 ((card‘ω) ≺ ω → ¬ (card‘ω) ≈ ω)
64, 5syl 17 . . 3 ((card‘ω) ∈ ω → ¬ (card‘ω) ≈ ω)
73, 6mt2 203 . 2 ¬ (card‘ω) ∈ ω
8 cardonle 9362 . . . 4 (ω ∈ On → (card‘ω) ⊆ ω)
91, 8ax-mp 5 . . 3 (card‘ω) ⊆ ω
10 cardon 9349 . . . 4 (card‘ω) ∈ On
1110, 1onsseli 6278 . . 3 ((card‘ω) ⊆ ω ↔ ((card‘ω) ∈ ω ∨ (card‘ω) = ω))
129, 11mpbi 233 . 2 ((card‘ω) ∈ ω ∨ (card‘ω) = ω)
137, 12mtpor 1772 1 (card‘ω) = ω
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 844   = wceq 1538  wcel 2115  wss 3910   class class class wbr 5039  Oncon0 6164  cfv 6328  ωcom 7555  cen 8481  csdm 8483  cardccrd 9340
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-inf2 9080
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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  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-om 7556  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-card 9344
This theorem is referenced by:  infxpidm2  9420  alephcard  9473  infenaleph  9494  alephval2  9971  pwfseqlem5  10062
  Copyright terms: Public domain W3C validator