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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 9893 | . . 3 β’ (π΄ β dom card β (cardβπ΄) = β© {π¦ β On β£ π¦ β π΄}) | |
2 | ssrab2 4038 | . . . 4 β’ {π¦ β On β£ π¦ β π΄} β On | |
3 | fvex 6856 | . . . . . 6 β’ (cardβπ΄) β V | |
4 | 1, 3 | eqeltrrdi 2843 | . . . . 5 β’ (π΄ β dom card β β© {π¦ β On β£ π¦ β π΄} β V) |
5 | intex 5295 | . . . . 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 7726 | . . . 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 588 | . . 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 2834 | . 2 β’ (π΄ β dom card β (cardβπ΄) β {π¦ β On β£ π¦ β π΄}) |
10 | breq1 5109 | . . . 4 β’ (π¦ = (cardβπ΄) β (π¦ β π΄ β (cardβπ΄) β π΄)) | |
11 | 10 | elrab 3646 | . . 3 β’ ((cardβπ΄) β {π¦ β On β£ π¦ β π΄} β ((cardβπ΄) β On β§ (cardβπ΄) β π΄)) |
12 | 11 | simprbi 498 | . 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 2107 β wne 2940 {crab 3406 Vcvv 3444 β wss 3911 β c0 4283 β© cint 4908 class class class wbr 5106 dom cdm 5634 Oncon0 6318 βcfv 6497 β cen 8883 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-pr 5385 |
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-rab 3407 df-v 3446 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-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-en 8887 df-card 9880 |
This theorem is referenced by: isnum3 9895 oncardid 9897 cardidm 9900 ficardom 9902 ficardid 9903 cardnn 9904 cardnueq0 9905 carden2a 9907 carden2b 9908 carddomi2 9911 sdomsdomcardi 9912 cardsdomelir 9914 cardsdomel 9915 infxpidm2 9958 dfac8b 9972 numdom 9979 alephnbtwn2 10013 alephsucdom 10020 infenaleph 10032 dfac12r 10087 cardadju 10135 pwsdompw 10145 cff1 10199 cfflb 10200 cflim2 10204 cfss 10206 cfslb 10207 domtriomlem 10383 cardid 10488 cardidg 10489 carden 10492 sdomsdomcard 10501 hargch 10614 gch2 10616 hashkf 14238 |
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