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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 8426 | . . . 4 ⊢ ω ∈ On | |
2 | oncardid 8665 | . . . 4 ⊢ (ω ∈ On → (card‘ω) ≈ ω) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (card‘ω) ≈ ω |
4 | nnsdom 8434 | . . . 4 ⊢ ((card‘ω) ∈ ω → (card‘ω) ≺ ω) | |
5 | sdomnen 7870 | . . . 4 ⊢ ((card‘ω) ≺ ω → ¬ (card‘ω) ≈ ω) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ ((card‘ω) ∈ ω → ¬ (card‘ω) ≈ ω) |
7 | 3, 6 | mt2 190 | . 2 ⊢ ¬ (card‘ω) ∈ ω |
8 | cardonle 8666 | . . . 4 ⊢ (ω ∈ On → (card‘ω) ⊆ ω) | |
9 | 1, 8 | ax-mp 5 | . . 3 ⊢ (card‘ω) ⊆ ω |
10 | cardon 8653 | . . . 4 ⊢ (card‘ω) ∈ On | |
11 | 10, 1 | onsseli 5759 | . . 3 ⊢ ((card‘ω) ⊆ ω ↔ ((card‘ω) ∈ ω ∨ (card‘ω) = ω)) |
12 | 9, 11 | mpbi 219 | . 2 ⊢ ((card‘ω) ∈ ω ∨ (card‘ω) = ω) |
13 | 7, 12 | mtpor 1686 | 1 ⊢ (card‘ω) = ω |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 382 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 class class class wbr 4583 Oncon0 5640 ‘cfv 5804 ωcom 6957 ≈ cen 7838 ≺ csdm 7840 cardccrd 8644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 |
This theorem is referenced by: infxpidm2 8723 alephcard 8776 infenaleph 8797 alephval2 9273 pwfseqlem5 9364 |
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