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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 9640 | . . . 4 ⊢ ω ∈ On | |
2 | oncardid 9950 | . . . 4 ⊢ (ω ∈ On → (card‘ω) ≈ ω) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (card‘ω) ≈ ω |
4 | nnsdom 9648 | . . . 4 ⊢ ((card‘ω) ∈ ω → (card‘ω) ≺ ω) | |
5 | sdomnen 8976 | . . . 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 9951 | . . . 4 ⊢ (ω ∈ On → (card‘ω) ⊆ ω) | |
9 | 1, 8 | ax-mp 5 | . . 3 ⊢ (card‘ω) ⊆ ω |
10 | cardon 9938 | . . . 4 ⊢ (card‘ω) ∈ On | |
11 | 10, 1 | onsseli 6478 | . . 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 1764 | 1 ⊢ (card‘ω) = ω |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ⊆ wss 3943 class class class wbr 5141 Oncon0 6357 ‘cfv 6536 ωcom 7851 ≈ cen 8935 ≺ csdm 8937 cardccrd 9929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-om 7852 df-1o 8464 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 |
This theorem is referenced by: infxpidm2 10011 alephcard 10064 infenaleph 10085 alephval2 10566 pwfseqlem5 10657 |
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