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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 8535 | . . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
2 | 1 | biimpi 218 | . 2 ⊢ (𝐴 ∈ Fin → ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) |
3 | finnum 9379 | . . . . . . . 8 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card) | |
4 | cardid2 9384 | . . . . . . . 8 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
5 | 3, 4 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ≈ 𝐴) |
6 | entr 8563 | . . . . . . 7 ⊢ (((card‘𝐴) ≈ 𝐴 ∧ 𝐴 ≈ 𝑥) → (card‘𝐴) ≈ 𝑥) | |
7 | 5, 6 | sylan 582 | . . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝑥) → (card‘𝐴) ≈ 𝑥) |
8 | cardon 9375 | . . . . . . 7 ⊢ (card‘𝐴) ∈ On | |
9 | onomeneq 8710 | . . . . . . 7 ⊢ (((card‘𝐴) ∈ On ∧ 𝑥 ∈ ω) → ((card‘𝐴) ≈ 𝑥 ↔ (card‘𝐴) = 𝑥)) | |
10 | 8, 9 | mpan 688 | . . . . . 6 ⊢ (𝑥 ∈ ω → ((card‘𝐴) ≈ 𝑥 ↔ (card‘𝐴) = 𝑥)) |
11 | 7, 10 | syl5ib 246 | . . . . 5 ⊢ (𝑥 ∈ ω → ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝑥) → (card‘𝐴) = 𝑥)) |
12 | eleq1a 2910 | . . . . 5 ⊢ (𝑥 ∈ ω → ((card‘𝐴) = 𝑥 → (card‘𝐴) ∈ ω)) | |
13 | 11, 12 | syld 47 | . . . 4 ⊢ (𝑥 ∈ ω → ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝑥) → (card‘𝐴) ∈ ω)) |
14 | 13 | expcomd 419 | . . 3 ⊢ (𝑥 ∈ ω → (𝐴 ≈ 𝑥 → (𝐴 ∈ Fin → (card‘𝐴) ∈ ω))) |
15 | 14 | rexlimiv 3282 | . 2 ⊢ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)) |
16 | 2, 15 | mpcom 38 | 1 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 class class class wbr 5068 dom cdm 5557 Oncon0 6193 ‘cfv 6357 ωcom 7582 ≈ cen 8508 Fincfn 8511 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 |
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: cardnn 9394 isinffi 9423 finnisoeu 9541 iunfictbso 9542 ficardun 9626 ficardun2 9627 pwsdompw 9628 ackbij1lem5 9648 ackbij1lem9 9652 ackbij1lem10 9653 ackbij1lem14 9657 ackbij1b 9663 ackbij2lem2 9664 ackbij2 9667 fin23lem22 9751 fin1a2lem11 9834 domtriomlem 9866 pwfseqlem4a 10085 pwfseqlem4 10086 hashkf 13695 hashginv 13697 hashcard 13719 hashcl 13720 hashdom 13743 hashun 13746 ishashinf 13824 ackbijnn 15185 mreexexd 16921 |
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