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Theorem cardom 9000
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 8714 . . . 4 ω ∈ On
2 oncardid 8970 . . . 4 (ω ∈ On → (card‘ω) ≈ ω)
31, 2ax-mp 5 . . 3 (card‘ω) ≈ ω
4 nnsdom 8722 . . . 4 ((card‘ω) ∈ ω → (card‘ω) ≺ ω)
5 sdomnen 8148 . . . 4 ((card‘ω) ≺ ω → ¬ (card‘ω) ≈ ω)
64, 5syl 17 . . 3 ((card‘ω) ∈ ω → ¬ (card‘ω) ≈ ω)
73, 6mt2 191 . 2 ¬ (card‘ω) ∈ ω
8 cardonle 8971 . . . 4 (ω ∈ On → (card‘ω) ⊆ ω)
91, 8ax-mp 5 . . 3 (card‘ω) ⊆ ω
10 cardon 8958 . . . 4 (card‘ω) ∈ On
1110, 1onsseli 6001 . . 3 ((card‘ω) ⊆ ω ↔ ((card‘ω) ∈ ω ∨ (card‘ω) = ω))
129, 11mpbi 220 . 2 ((card‘ω) ∈ ω ∨ (card‘ω) = ω)
137, 12mtpor 1842 1 (card‘ω) = ω
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 382   = wceq 1630  wcel 2137  wss 3713   class class class wbr 4802  Oncon0 5882  cfv 6047  ωcom 7228  cen 8116  csdm 8118  cardccrd 8949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-inf2 8709
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-om 7229  df-er 7909  df-en 8120  df-dom 8121  df-sdom 8122  df-fin 8123  df-card 8953
This theorem is referenced by:  infxpidm2  9028  alephcard  9081  infenaleph  9102  alephval2  9584  pwfseqlem5  9675
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