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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 8919 | . . 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 9889 | . . . . . . . 8 β’ (π΄ β Fin β π΄ β dom card) | |
4 | cardid2 9894 | . . . . . . . 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 8949 | . . . . . . 7 β’ (((cardβπ΄) β π΄ β§ π΄ β π₯) β (cardβπ΄) β π₯) | |
7 | 5, 6 | sylan 581 | . . . . . 6 β’ ((π΄ β Fin β§ π΄ β π₯) β (cardβπ΄) β π₯) |
8 | cardon 9885 | . . . . . . 7 β’ (cardβπ΄) β On | |
9 | onomeneq 9175 | . . . . . . 7 β’ (((cardβπ΄) β On β§ π₯ β Ο) β ((cardβπ΄) β π₯ β (cardβπ΄) = π₯)) | |
10 | 8, 9 | mpan 689 | . . . . . 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 2829 | . . . . 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 418 | . . 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 397 = wceq 1542 β wcel 2107 βwrex 3070 class class class wbr 5106 dom cdm 5634 Oncon0 6318 βcfv 6497 Οcom 7803 β cen 8883 Fincfn 8886 cardccrd 9876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-om 7804 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-card 9880 |
This theorem is referenced by: cardnn 9904 isinffi 9933 finnisoeu 10054 iunfictbso 10055 ficardadju 10140 ficardun 10141 ficardunOLD 10142 ficardun2 10143 ficardun2OLD 10144 pwsdompw 10145 ackbij1lem5 10165 ackbij1lem9 10169 ackbij1lem10 10170 ackbij1lem14 10174 ackbij1b 10180 ackbij2lem2 10181 ackbij2 10184 fin23lem22 10268 fin1a2lem11 10351 domtriomlem 10383 pwfseqlem4a 10602 pwfseqlem4 10603 hashkf 14238 hashginv 14240 hashcard 14261 hashcl 14262 hashdom 14285 hashun 14288 ishashinf 14368 ackbijnn 15718 mreexexd 17533 |
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