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Mirrors > Home > MPE Home > Th. List > cardom | Structured version Visualization version GIF version |
Description: The set of natural numbers is a cardinal number. Theorem 18.11 of [Monk1] p. 133. (Contributed by NM, 28-Oct-2003.) |
Ref | Expression |
---|---|
cardom | ⊢ (card‘ω) = ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omelon 9587 | . . . 4 ⊢ ω ∈ On | |
2 | oncardid 9897 | . . . 4 ⊢ (ω ∈ On → (card‘ω) ≈ ω) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (card‘ω) ≈ ω |
4 | nnsdom 9595 | . . . 4 ⊢ ((card‘ω) ∈ ω → (card‘ω) ≺ ω) | |
5 | sdomnen 8924 | . . . 4 ⊢ ((card‘ω) ≺ ω → ¬ (card‘ω) ≈ ω) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ ((card‘ω) ∈ ω → ¬ (card‘ω) ≈ ω) |
7 | 3, 6 | mt2 199 | . 2 ⊢ ¬ (card‘ω) ∈ ω |
8 | cardonle 9898 | . . . 4 ⊢ (ω ∈ On → (card‘ω) ⊆ ω) | |
9 | 1, 8 | ax-mp 5 | . . 3 ⊢ (card‘ω) ⊆ ω |
10 | cardon 9885 | . . . 4 ⊢ (card‘ω) ∈ On | |
11 | 10, 1 | onsseli 6439 | . . 3 ⊢ ((card‘ω) ⊆ ω ↔ ((card‘ω) ∈ ω ∨ (card‘ω) = ω)) |
12 | 9, 11 | mpbi 229 | . 2 ⊢ ((card‘ω) ∈ ω ∨ (card‘ω) = ω) |
13 | 7, 12 | mtpor 1773 | 1 ⊢ (card‘ω) = ω |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ⊆ wss 3911 class class class wbr 5106 Oncon0 6318 ‘cfv 6497 ωcom 7803 ≈ cen 8883 ≺ csdm 8885 cardccrd 9876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-om 7804 df-1o 8413 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-card 9880 |
This theorem is referenced by: infxpidm2 9958 alephcard 10011 infenaleph 10032 alephval2 10513 pwfseqlem5 10604 |
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