| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 9686 | . . . 4 ⊢ ω ∈ On | |
| 2 | oncardid 9996 | . . . 4 ⊢ (ω ∈ On → (card‘ω) ≈ ω) | |
| 3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (card‘ω) ≈ ω |
| 4 | nnsdom 9694 | . . . 4 ⊢ ((card‘ω) ∈ ω → (card‘ω) ≺ ω) | |
| 5 | sdomnen 9021 | . . . 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 9997 | . . . 4 ⊢ (ω ∈ On → (card‘ω) ⊆ ω) | |
| 9 | 1, 8 | ax-mp 5 | . . 3 ⊢ (card‘ω) ⊆ ω |
| 10 | cardon 9984 | . . . 4 ⊢ (card‘ω) ∈ On | |
| 11 | 10, 1 | onsseli 6505 | . . 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 1770 | 1 ⊢ (card‘ω) = ω |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 class class class wbr 5143 Oncon0 6384 ‘cfv 6561 ωcom 7887 ≈ cen 8982 ≺ csdm 8984 cardccrd 9975 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-1o 8506 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 |
| This theorem is referenced by: infxpidm2 10057 alephcard 10110 infenaleph 10131 alephval2 10612 pwfseqlem5 10703 |
| Copyright terms: Public domain | W3C validator |