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Theorem cardom 9126
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 8821 . . . 4 ω ∈ On
2 oncardid 9096 . . . 4 (ω ∈ On → (card‘ω) ≈ ω)
31, 2ax-mp 5 . . 3 (card‘ω) ≈ ω
4 nnsdom 8829 . . . 4 ((card‘ω) ∈ ω → (card‘ω) ≺ ω)
5 sdomnen 8252 . . . 4 ((card‘ω) ≺ ω → ¬ (card‘ω) ≈ ω)
64, 5syl 17 . . 3 ((card‘ω) ∈ ω → ¬ (card‘ω) ≈ ω)
73, 6mt2 192 . 2 ¬ (card‘ω) ∈ ω
8 cardonle 9097 . . . 4 (ω ∈ On → (card‘ω) ⊆ ω)
91, 8ax-mp 5 . . 3 (card‘ω) ⊆ ω
10 cardon 9084 . . . 4 (card‘ω) ∈ On
1110, 1onsseli 6078 . . 3 ((card‘ω) ⊆ ω ↔ ((card‘ω) ∈ ω ∨ (card‘ω) = ω))
129, 11mpbi 222 . 2 ((card‘ω) ∈ ω ∨ (card‘ω) = ω)
137, 12mtpor 1871 1 (card‘ω) = ω
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 880   = wceq 1658  wcel 2166  wss 3799   class class class wbr 4874  Oncon0 5964  cfv 6124  ωcom 7327  cen 8220  csdm 8222  cardccrd 9075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-inf2 8816
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-int 4699  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-om 7328  df-er 8010  df-en 8224  df-dom 8225  df-sdom 8226  df-fin 8227  df-card 9079
This theorem is referenced by:  infxpidm2  9154  alephcard  9207  infenaleph  9228  alephval2  9710  pwfseqlem5  9801
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