Theorem cardom 9879

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

Step	Hyp	Ref	Expression
1		omelon 9536	. . . 4 ⊢ ω ∈ On
2		oncardid 9849	. . . 4 ⊢ (ω ∈ On → (card‘ω) ≈ ω)
3	1, 2	ax-mp 5	. . 3 ⊢ (card‘ω) ≈ ω
4		nnsdom 9544	. . . 4 ⊢ ((card‘ω) ∈ ω → (card‘ω) ≺ ω)
5		sdomnen 8903	. . . 4 ⊢ ((card‘ω) ≺ ω → ¬ (card‘ω) ≈ ω)
6	4, 5	syl 17	. . 3 ⊢ ((card‘ω) ∈ ω → ¬ (card‘ω) ≈ ω)
7	3, 6	mt2 200	. 2 ⊢ ¬ (card‘ω) ∈ ω
8		cardonle 9850	. . . 4 ⊢ (ω ∈ On → (card‘ω) ⊆ ω)
9	1, 8	ax-mp 5	. . 3 ⊢ (card‘ω) ⊆ ω
10		cardon 9837	. . . 4 ⊢ (card‘ω) ∈ On
11	10, 1	onsseli 6428	. . 3 ⊢ ((card‘ω) ⊆ ω ↔ ((card‘ω) ∈ ω ∨ (card‘ω) = ω))
12	9, 11	mpbi 230	. 2 ⊢ ((card‘ω) ∈ ω ∨ (card‘ω) = ω)
13	7, 12	mtpor 1771	1 ⊢ (card‘ω) = ω

Syntax hints:  ¬ wn 3   ∨ wo 847   = wceq 1541   ∈ wcel 2111   ⊆ wss 3897   class class class wbr 5089  Oncon0 6306  ‘cfv 6481  ωcom 7796   ≈ cen 8866   ≺ csdm 8868  cardccrd 9828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-inf2 9531

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-om 7797  df-1o 8385  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-card 9832

This theorem is referenced by:  infxpidm2  9908  alephcard  9961  infenaleph  9982  alephval2  10463  pwfseqlem5  10554