Theorem cardom 3632

Description: The set of natural numbers is a cardinal number. Theorem 18.11 of [Monk1] p. 133.

Assertion

Ref	Expression
cardom	⊢ (card ‘ω) = ω

Proof of Theorem cardom

Step	Hyp	Ref	Expression
1		omelon 3476	. . . 4 ⊢ ω ∈ On
2		oncardid 3628	. . . 4 ⊢ (ω ∈ On → (card ‘ω) ≈ ω)
3	1, 2	ax-mp 6	. . 3 ⊢ (card ‘ω) ≈ ω
4		nnsdom 3481	. . . 4 ⊢ ((card ‘ω) ∈ ω → (card ‘ω) ≺ ω)
5		sdomnen 3291	. . . 4 ⊢ ((card ‘ω) ≺ ω → ¬ (card ‘ω) ≈ ω)
6	4, 5	syl 12	. . 3 ⊢ ((card ‘ω) ∈ ω → ¬ (card ‘ω) ≈ ω)
7	3, 6	mt2 96	. 2 ⊢ ¬ (card ‘ω) ∈ ω
8		cardonle 3629	. . . . 5 ⊢ (ω ∈ On → (card ‘ω) ⊆ ω)
9	1, 8	ax-mp 6	. . . 4 ⊢ (card ‘ω) ⊆ ω
10		oncardon 3627	. . . . . 6 ⊢ (ω ∈ On → (card ‘ω) ∈ On)
11	1, 10	ax-mp 6	. . . . 5 ⊢ (card ‘ω) ∈ On
12	11, 1	onssel 2357	. . . 4 ⊢ ((card ‘ω) ⊆ ω ↔ ((card ‘ω) ∈ ω ∨ (card ‘ω) = ω))
13	9, 12	mpbi 164	. . 3 ⊢ ((card ‘ω) ∈ ω ∨ (card ‘ω) = ω)
14	13	ori 200	. 2 ⊢ (¬ (card ‘ω) ∈ ω → (card ‘ω) = ω)
15	7, 14	ax-mp 6	1 ⊢ (card ‘ω) = ω

Syntax hints:  ¬ wn 1   ∨ wo 195   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487   class class class wbr 2054  Oncon0 2199  ωcom 2372   ‘cfv 2422   ≈ cen 3271   ≺ csdm 3273  cardccrd 3620

This theorem is referenced by:  alephcard 3673  cfom 3710

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-er 3200  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623

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