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Mirrors > Home > MPE Home > Th. List > oncardval | Structured version Visualization version GIF version |
Description: The value of the cardinal number function with an ordinal number as its argument. Unlike cardval 10569, this theorem does not require the Axiom of Choice. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.) |
Ref | Expression |
---|---|
oncardval | β’ (π΄ β On β (cardβπ΄) = β© {π₯ β On β£ π₯ β π΄}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onenon 9972 | . 2 β’ (π΄ β On β π΄ β dom card) | |
2 | cardval3 9975 | . 2 β’ (π΄ β dom card β (cardβπ΄) = β© {π₯ β On β£ π₯ β π΄}) | |
3 | 1, 2 | syl 17 | 1 β’ (π΄ β On 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 {crab 3419 β© cint 4944 class class class wbr 5143 dom cdm 5672 Oncon0 6364 βcfv 6543 β cen 8959 cardccrd 9958 |
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 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6367 df-on 6368 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-en 8963 df-card 9962 |
This theorem is referenced by: cardonle 9980 cardidm 9982 iscard2 9999 |
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