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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 9684 | . . . 4 ⊢ ω ∈ On | |
2 | oncardid 9994 | . . . 4 ⊢ (ω ∈ On → (card‘ω) ≈ ω) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (card‘ω) ≈ ω |
4 | nnsdom 9692 | . . . 4 ⊢ ((card‘ω) ∈ ω → (card‘ω) ≺ ω) | |
5 | sdomnen 9020 | . . . 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 9995 | . . . 4 ⊢ (ω ∈ On → (card‘ω) ⊆ ω) | |
9 | 1, 8 | ax-mp 5 | . . 3 ⊢ (card‘ω) ⊆ ω |
10 | cardon 9982 | . . . 4 ⊢ (card‘ω) ∈ On | |
11 | 10, 1 | onsseli 6507 | . . 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 1767 | 1 ⊢ (card‘ω) = ω |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 class class class wbr 5148 Oncon0 6386 ‘cfv 6563 ωcom 7887 ≈ cen 8981 ≺ csdm 8983 cardccrd 9973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-om 7888 df-1o 8505 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 |
This theorem is referenced by: infxpidm2 10055 alephcard 10108 infenaleph 10129 alephval2 10610 pwfseqlem5 10701 |
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