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 9111 | . . . 4 ⊢ ω ∈ On | |
2 | oncardid 9387 | . . . 4 ⊢ (ω ∈ On → (card‘ω) ≈ ω) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (card‘ω) ≈ ω |
4 | nnsdom 9119 | . . . 4 ⊢ ((card‘ω) ∈ ω → (card‘ω) ≺ ω) | |
5 | sdomnen 8540 | . . . 4 ⊢ ((card‘ω) ≺ ω → ¬ (card‘ω) ≈ ω) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ ((card‘ω) ∈ ω → ¬ (card‘ω) ≈ ω) |
7 | 3, 6 | mt2 202 | . 2 ⊢ ¬ (card‘ω) ∈ ω |
8 | cardonle 9388 | . . . 4 ⊢ (ω ∈ On → (card‘ω) ⊆ ω) | |
9 | 1, 8 | ax-mp 5 | . . 3 ⊢ (card‘ω) ⊆ ω |
10 | cardon 9375 | . . . 4 ⊢ (card‘ω) ∈ On | |
11 | 10, 1 | onsseli 6307 | . . 3 ⊢ ((card‘ω) ⊆ ω ↔ ((card‘ω) ∈ ω ∨ (card‘ω) = ω)) |
12 | 9, 11 | mpbi 232 | . 2 ⊢ ((card‘ω) ∈ ω ∨ (card‘ω) = ω) |
13 | 7, 12 | mtpor 1771 | 1 ⊢ (card‘ω) = ω |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 class class class wbr 5068 Oncon0 6193 ‘cfv 6357 ωcom 7582 ≈ cen 8508 ≺ csdm 8510 cardccrd 9366 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-om 7583 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 |
This theorem is referenced by: infxpidm2 9445 alephcard 9498 infenaleph 9519 alephval2 9996 pwfseqlem5 10087 |
Copyright terms: Public domain | W3C validator |