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Mirrors > Home > MPE Home > Th. List > cardval3 | Structured version Visualization version GIF version |
Description: An alternate definition of the value of (cardβπ΄) that does not require AC to prove. (Contributed by Mario Carneiro, 7-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.) |
Ref | Expression |
---|---|
cardval3 | β’ (π΄ β dom card β (cardβπ΄) = β© {π₯ β On β£ π₯ β π΄}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3492 | . 2 β’ (π΄ β dom card β π΄ β V) | |
2 | isnum2 9976 | . . . 4 β’ (π΄ β dom card β βπ₯ β On π₯ β π΄) | |
3 | rabn0 4389 | . . . 4 β’ ({π₯ β On β£ π₯ β π΄} β β β βπ₯ β On π₯ β π΄) | |
4 | intex 5343 | . . . 4 β’ ({π₯ β On β£ π₯ β π΄} β β β β© {π₯ β On β£ π₯ β π΄} β V) | |
5 | 2, 3, 4 | 3bitr2i 298 | . . 3 β’ (π΄ β dom card β β© {π₯ β On β£ π₯ β π΄} β V) |
6 | 5 | biimpi 215 | . 2 β’ (π΄ β dom card β β© {π₯ β On β£ π₯ β π΄} β V) |
7 | breq2 5156 | . . . . 5 β’ (π¦ = π΄ β (π₯ β π¦ β π₯ β π΄)) | |
8 | 7 | rabbidv 3438 | . . . 4 β’ (π¦ = π΄ β {π₯ β On β£ π₯ β π¦} = {π₯ β On β£ π₯ β π΄}) |
9 | 8 | inteqd 4958 | . . 3 β’ (π¦ = π΄ β β© {π₯ β On β£ π₯ β π¦} = β© {π₯ β On β£ π₯ β π΄}) |
10 | df-card 9970 | . . 3 β’ card = (π¦ β V β¦ β© {π₯ β On β£ π₯ β π¦}) | |
11 | 9, 10 | fvmptg 7008 | . 2 β’ ((π΄ β V β§ β© {π₯ β On β£ π₯ β π΄} β V) β (cardβπ΄) = β© {π₯ β On β£ π₯ β π΄}) |
12 | 1, 6, 11 | syl2anc 582 | 1 β’ (π΄ β dom card β (cardβπ΄) = β© {π₯ β On β£ π₯ β π΄}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wne 2937 βwrex 3067 {crab 3430 Vcvv 3473 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: cardid2 9984 oncardval 9986 cardidm 9990 cardne 9996 cardval 10577 |
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