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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 3466 | . 2 β’ (π΄ β dom card β π΄ β V) | |
2 | isnum2 9888 | . . . 4 β’ (π΄ β dom card β βπ₯ β On π₯ β π΄) | |
3 | rabn0 4350 | . . . 4 β’ ({π₯ β On β£ π₯ β π΄} β β β βπ₯ β On π₯ β π΄) | |
4 | intex 5299 | . . . 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 299 | . . 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 5114 | . . . . 5 β’ (π¦ = π΄ β (π₯ β π¦ β π₯ β π΄)) | |
8 | 7 | rabbidv 3418 | . . . 4 β’ (π¦ = π΄ β {π₯ β On β£ π₯ β π¦} = {π₯ β On β£ π₯ β π΄}) |
9 | 8 | inteqd 4917 | . . 3 β’ (π¦ = π΄ β β© {π₯ β On β£ π₯ β π¦} = β© {π₯ β On β£ π₯ β π΄}) |
10 | df-card 9882 | . . 3 β’ card = (π¦ β V β¦ β© {π₯ β On β£ π₯ β π¦}) | |
11 | 9, 10 | fvmptg 6951 | . 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 585 | 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 1542 β wcel 2107 β wne 2944 βwrex 3074 {crab 3410 Vcvv 3448 β c0 4287 β© cint 4912 class class class wbr 5110 dom cdm 5638 Oncon0 6322 βcfv 6501 β cen 8887 cardccrd 9878 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-ord 6325 df-on 6326 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-fv 6509 df-en 8891 df-card 9882 |
This theorem is referenced by: cardid2 9896 oncardval 9898 cardidm 9902 cardne 9908 cardval 10489 |
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