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Mirrors > Home > MPE Home > Th. List > cardid2 | Structured version Visualization version GIF version |
Description: Any numerable set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.) |
Ref | Expression |
---|---|
cardid2 | β’ (π΄ β dom card β (cardβπ΄) β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardval3 9983 | . . 3 β’ (π΄ β dom card β (cardβπ΄) = β© {π¦ β On β£ π¦ β π΄}) | |
2 | ssrab2 4077 | . . . 4 β’ {π¦ β On β£ π¦ β π΄} β On | |
3 | fvex 6915 | . . . . . 6 β’ (cardβπ΄) β V | |
4 | 1, 3 | eqeltrrdi 2838 | . . . . 5 β’ (π΄ β dom card β β© {π¦ β On β£ π¦ β π΄} β V) |
5 | intex 5343 | . . . . 5 β’ ({π¦ β On β£ π¦ β π΄} β β β β© {π¦ β On β£ π¦ β π΄} β V) | |
6 | 4, 5 | sylibr 233 | . . . 4 β’ (π΄ β dom card β {π¦ β On β£ π¦ β π΄} β β ) |
7 | onint 7799 | . . . 4 β’ (({π¦ β On β£ π¦ β π΄} β On β§ {π¦ β On β£ π¦ β π΄} β β ) β β© {π¦ β On β£ π¦ β π΄} β {π¦ β On β£ π¦ β π΄}) | |
8 | 2, 6, 7 | sylancr 585 | . . 3 β’ (π΄ β dom card β β© {π¦ β On β£ π¦ β π΄} β {π¦ β On β£ π¦ β π΄}) |
9 | 1, 8 | eqeltrd 2829 | . 2 β’ (π΄ β dom card β (cardβπ΄) β {π¦ β On β£ π¦ β π΄}) |
10 | breq1 5155 | . . . 4 β’ (π¦ = (cardβπ΄) β (π¦ β π΄ β (cardβπ΄) β π΄)) | |
11 | 10 | elrab 3684 | . . 3 β’ ((cardβπ΄) β {π¦ β On β£ π¦ β π΄} β ((cardβπ΄) β On β§ (cardβπ΄) β π΄)) |
12 | 11 | simprbi 495 | . 2 β’ ((cardβπ΄) β {π¦ β On β£ π¦ β π΄} β (cardβπ΄) β π΄) |
13 | 9, 12 | syl 17 | 1 β’ (π΄ β dom card β (cardβπ΄) β π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2098 β wne 2937 {crab 3430 Vcvv 3473 β wss 3949 β c0 4326 β© cint 4953 class class class wbr 5152 dom cdm 5682 Oncon0 6374 βcfv 6553 β cen 8967 cardccrd 9966 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-en 8971 df-card 9970 |
This theorem is referenced by: isnum3 9985 oncardid 9987 cardidm 9990 ficardom 9992 ficardid 9993 cardnn 9994 cardnueq0 9995 carden2a 9997 carden2b 9998 carddomi2 10001 sdomsdomcardi 10002 cardsdomelir 10004 cardsdomel 10005 infxpidm2 10048 dfac8b 10062 numdom 10069 alephnbtwn2 10103 alephsucdom 10110 infenaleph 10122 dfac12r 10177 cardadju 10225 pwsdompw 10235 cff1 10289 cfflb 10290 cflim2 10294 cfss 10296 cfslb 10297 domtriomlem 10473 cardid 10578 cardidg 10579 carden 10582 sdomsdomcard 10591 hargch 10704 gch2 10706 hashkf 14331 |
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