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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 9946 | . . 3 β’ (π΄ β dom card β (cardβπ΄) = β© {π¦ β On β£ π¦ β π΄}) | |
2 | ssrab2 4072 | . . . 4 β’ {π¦ β On β£ π¦ β π΄} β On | |
3 | fvex 6897 | . . . . . 6 β’ (cardβπ΄) β V | |
4 | 1, 3 | eqeltrrdi 2836 | . . . . 5 β’ (π΄ β dom card β β© {π¦ β On β£ π¦ β π΄} β V) |
5 | intex 5330 | . . . . 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 7774 | . . . 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 586 | . . 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 2827 | . 2 β’ (π΄ β dom card β (cardβπ΄) β {π¦ β On β£ π¦ β π΄}) |
10 | breq1 5144 | . . . 4 β’ (π¦ = (cardβπ΄) β (π¦ β π΄ β (cardβπ΄) β π΄)) | |
11 | 10 | elrab 3678 | . . 3 β’ ((cardβπ΄) β {π¦ β On β£ π¦ β π΄} β ((cardβπ΄) β On β§ (cardβπ΄) β π΄)) |
12 | 11 | simprbi 496 | . 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 2934 {crab 3426 Vcvv 3468 β wss 3943 β c0 4317 β© cint 4943 class class class wbr 5141 dom cdm 5669 Oncon0 6357 βcfv 6536 β cen 8935 cardccrd 9929 |
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-pr 5420 |
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-rab 3427 df-v 3470 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 6360 df-on 6361 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-en 8939 df-card 9933 |
This theorem is referenced by: isnum3 9948 oncardid 9950 cardidm 9953 ficardom 9955 ficardid 9956 cardnn 9957 cardnueq0 9958 carden2a 9960 carden2b 9961 carddomi2 9964 sdomsdomcardi 9965 cardsdomelir 9967 cardsdomel 9968 infxpidm2 10011 dfac8b 10025 numdom 10032 alephnbtwn2 10066 alephsucdom 10073 infenaleph 10085 dfac12r 10140 cardadju 10188 pwsdompw 10198 cff1 10252 cfflb 10253 cflim2 10257 cfss 10259 cfslb 10260 domtriomlem 10436 cardid 10541 cardidg 10542 carden 10545 sdomsdomcard 10554 hargch 10667 gch2 10669 hashkf 14294 |
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