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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 9949 | . . 3 β’ (π΄ β dom card β (cardβπ΄) = β© {π¦ β On β£ π¦ β π΄}) | |
2 | ssrab2 4076 | . . . 4 β’ {π¦ β On β£ π¦ β π΄} β On | |
3 | fvex 6903 | . . . . . 6 β’ (cardβπ΄) β V | |
4 | 1, 3 | eqeltrrdi 2840 | . . . . 5 β’ (π΄ β dom card β β© {π¦ β On β£ π¦ β π΄} β V) |
5 | intex 5336 | . . . . 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 7780 | . . . 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 2831 | . 2 β’ (π΄ β dom card β (cardβπ΄) β {π¦ β On β£ π¦ β π΄}) |
10 | breq1 5150 | . . . 4 β’ (π¦ = (cardβπ΄) β (π¦ β π΄ β (cardβπ΄) β π΄)) | |
11 | 10 | elrab 3682 | . . 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 2104 β wne 2938 {crab 3430 Vcvv 3472 β wss 3947 β c0 4321 β© cint 4949 class class class wbr 5147 dom cdm 5675 Oncon0 6363 βcfv 6542 β cen 8938 cardccrd 9932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6366 df-on 6367 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-en 8942 df-card 9936 |
This theorem is referenced by: isnum3 9951 oncardid 9953 cardidm 9956 ficardom 9958 ficardid 9959 cardnn 9960 cardnueq0 9961 carden2a 9963 carden2b 9964 carddomi2 9967 sdomsdomcardi 9968 cardsdomelir 9970 cardsdomel 9971 infxpidm2 10014 dfac8b 10028 numdom 10035 alephnbtwn2 10069 alephsucdom 10076 infenaleph 10088 dfac12r 10143 cardadju 10191 pwsdompw 10201 cff1 10255 cfflb 10256 cflim2 10260 cfss 10262 cfslb 10263 domtriomlem 10439 cardid 10544 cardidg 10545 carden 10548 sdomsdomcard 10557 hargch 10670 gch2 10672 hashkf 14296 |
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