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Theorem cardom 10016
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 9676 . . . 4 ω ∈ On
2 oncardid 9986 . . . 4 (ω ∈ On → (card‘ω) ≈ ω)
31, 2ax-mp 5 . . 3 (card‘ω) ≈ ω
4 nnsdom 9684 . . . 4 ((card‘ω) ∈ ω → (card‘ω) ≺ ω)
5 sdomnen 9002 . . . 4 ((card‘ω) ≺ ω → ¬ (card‘ω) ≈ ω)
64, 5syl 17 . . 3 ((card‘ω) ∈ ω → ¬ (card‘ω) ≈ ω)
73, 6mt2 199 . 2 ¬ (card‘ω) ∈ ω
8 cardonle 9987 . . . 4 (ω ∈ On → (card‘ω) ⊆ ω)
91, 8ax-mp 5 . . 3 (card‘ω) ⊆ ω
10 cardon 9974 . . . 4 (card‘ω) ∈ On
1110, 1onsseli 6492 . . 3 ((card‘ω) ⊆ ω ↔ ((card‘ω) ∈ ω ∨ (card‘ω) = ω))
129, 11mpbi 229 . 2 ((card‘ω) ∈ ω ∨ (card‘ω) = ω)
137, 12mtpor 1764 1 (card‘ω) = ω
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 845   = wceq 1533  wcel 2098  wss 3944   class class class wbr 5149  Oncon0 6371  cfv 6549  ωcom 7871  cen 8961  csdm 8963  cardccrd 9965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9671
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-om 7872  df-1o 8487  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-card 9969
This theorem is referenced by:  infxpidm2  10047  alephcard  10100  infenaleph  10121  alephval2  10602  pwfseqlem5  10693
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