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Mirrors > Home > MPE Home > Th. List > ficardom | Structured version Visualization version GIF version |
Description: The cardinal number of a finite set is a finite ordinal. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 4-Feb-2013.) |
Ref | Expression |
---|---|
ficardom | ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 7865 | . . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
2 | 1 | biimpi 205 | . 2 ⊢ (𝐴 ∈ Fin → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
3 | finnum 8657 | . . . . . . . 8 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card) | |
4 | cardid2 8662 | . . . . . . . 8 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
5 | 3, 4 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ≈ 𝐴) |
6 | entr 7894 | . . . . . . 7 ⊢ (((card‘𝐴) ≈ 𝐴 ∧ 𝐴 ≈ 𝑥) → (card‘𝐴) ≈ 𝑥) | |
7 | 5, 6 | sylan 487 | . . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝑥) → (card‘𝐴) ≈ 𝑥) |
8 | cardon 8653 | . . . . . . 7 ⊢ (card‘𝐴) ∈ On | |
9 | onomeneq 8035 | . . . . . . 7 ⊢ (((card‘𝐴) ∈ On ∧ 𝑥 ∈ ω) → ((card‘𝐴) ≈ 𝑥 ↔ (card‘𝐴) = 𝑥)) | |
10 | 8, 9 | mpan 702 | . . . . . 6 ⊢ (𝑥 ∈ ω → ((card‘𝐴) ≈ 𝑥 ↔ (card‘𝐴) = 𝑥)) |
11 | 7, 10 | syl5ib 233 | . . . . 5 ⊢ (𝑥 ∈ ω → ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝑥) → (card‘𝐴) = 𝑥)) |
12 | eleq1a 2683 | . . . . 5 ⊢ (𝑥 ∈ ω → ((card‘𝐴) = 𝑥 → (card‘𝐴) ∈ ω)) | |
13 | 11, 12 | syld 46 | . . . 4 ⊢ (𝑥 ∈ ω → ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝑥) → (card‘𝐴) ∈ ω)) |
14 | 13 | expcomd 453 | . . 3 ⊢ (𝑥 ∈ ω → (𝐴 ≈ 𝑥 → (𝐴 ∈ Fin → (card‘𝐴) ∈ ω))) |
15 | 14 | rexlimiv 3009 | . 2 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)) |
16 | 2, 15 | mpcom 37 | 1 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 class class class wbr 4583 dom cdm 5038 Oncon0 5640 ‘cfv 5804 ωcom 6957 ≈ cen 7838 Fincfn 7841 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 |
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: cardnn 8672 isinffi 8701 finnisoeu 8819 iunfictbso 8820 ficardun 8907 ficardun2 8908 pwsdompw 8909 ackbij1lem5 8929 ackbij1lem9 8933 ackbij1lem10 8934 ackbij1lem14 8938 ackbij1b 8944 ackbij2lem2 8945 ackbij2 8948 fin23lem22 9032 fin1a2lem11 9115 domtriomlem 9147 pwfseqlem4a 9362 pwfseqlem4 9363 hashkf 12981 hashginv 12983 hashcard 13008 hashcl 13009 hashdom 13029 hashun 13032 ishashinf 13104 ackbijnn 14399 mreexexd 16131 mreexexdOLD 16132 |
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