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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 | β’ (π΄ β Fin β (cardβπ΄) β Ο) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 8974 | . . 3 β’ (π΄ β Fin β βπ₯ β Ο π΄ β π₯) | |
2 | 1 | biimpi 215 | . 2 β’ (π΄ β Fin β βπ₯ β Ο π΄ β π₯) |
3 | finnum 9945 | . . . . . . . 8 β’ (π΄ β Fin β π΄ β dom card) | |
4 | cardid2 9950 | . . . . . . . 8 β’ (π΄ β dom card β (cardβπ΄) β π΄) | |
5 | 3, 4 | syl 17 | . . . . . . 7 β’ (π΄ β Fin β (cardβπ΄) β π΄) |
6 | entr 9004 | . . . . . . 7 β’ (((cardβπ΄) β π΄ β§ π΄ β π₯) β (cardβπ΄) β π₯) | |
7 | 5, 6 | sylan 579 | . . . . . 6 β’ ((π΄ β Fin β§ π΄ β π₯) β (cardβπ΄) β π₯) |
8 | cardon 9941 | . . . . . . 7 β’ (cardβπ΄) β On | |
9 | onomeneq 9230 | . . . . . . 7 β’ (((cardβπ΄) β On β§ π₯ β Ο) β ((cardβπ΄) β π₯ β (cardβπ΄) = π₯)) | |
10 | 8, 9 | mpan 687 | . . . . . 6 β’ (π₯ β Ο β ((cardβπ΄) β π₯ β (cardβπ΄) = π₯)) |
11 | 7, 10 | imbitrid 243 | . . . . 5 β’ (π₯ β Ο β ((π΄ β Fin β§ π΄ β π₯) β (cardβπ΄) = π₯)) |
12 | eleq1a 2822 | . . . . 5 β’ (π₯ β Ο β ((cardβπ΄) = π₯ β (cardβπ΄) β Ο)) | |
13 | 11, 12 | syld 47 | . . . 4 β’ (π₯ β Ο β ((π΄ β Fin β§ π΄ β π₯) β (cardβπ΄) β Ο)) |
14 | 13 | expcomd 416 | . . 3 β’ (π₯ β Ο β (π΄ β π₯ β (π΄ β Fin β (cardβπ΄) β Ο))) |
15 | 14 | rexlimiv 3142 | . 2 β’ (βπ₯ β Ο π΄ β π₯ β (π΄ β Fin β (cardβπ΄) β Ο)) |
16 | 2, 15 | mpcom 38 | 1 β’ (π΄ β Fin β (cardβπ΄) β Ο) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 βwrex 3064 class class class wbr 5141 dom cdm 5669 Oncon0 6358 βcfv 6537 Οcom 7852 β cen 8938 Fincfn 8941 cardccrd 9932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-om 7853 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 |
This theorem is referenced by: cardnn 9960 isinffi 9989 finnisoeu 10110 iunfictbso 10111 ficardadju 10196 ficardun 10197 ficardunOLD 10198 ficardun2 10199 ficardun2OLD 10200 pwsdompw 10201 ackbij1lem5 10221 ackbij1lem9 10225 ackbij1lem10 10226 ackbij1lem14 10230 ackbij1b 10236 ackbij2lem2 10237 ackbij2 10240 fin23lem22 10324 fin1a2lem11 10407 domtriomlem 10439 pwfseqlem4a 10658 pwfseqlem4 10659 hashkf 14297 hashginv 14299 hashcard 14320 hashcl 14321 hashdom 14344 hashun 14347 ishashinf 14430 ackbijnn 15780 mreexexd 17601 |
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