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Mirrors > Home > MPE Home > Th. List > cardval2 | Structured version Visualization version GIF version |
Description: An alternate version of the value of the cardinal number of a set. Compare cardval 10543. This theorem could be used to give a simpler definition of card in place of df-card 9936. It apparently does not occur in the literature. (Contributed by NM, 7-Nov-2003.) |
Ref | Expression |
---|---|
cardval2 | β’ (π΄ β dom card β (cardβπ΄) = {π₯ β On β£ π₯ βΊ π΄}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardon 9941 | . . . . . 6 β’ (cardβπ΄) β On | |
2 | 1 | oneli 6472 | . . . . 5 β’ (π₯ β (cardβπ΄) β π₯ β On) |
3 | 2 | pm4.71ri 560 | . . . 4 β’ (π₯ β (cardβπ΄) β (π₯ β On β§ π₯ β (cardβπ΄))) |
4 | cardsdomel 9971 | . . . . . 6 β’ ((π₯ β On β§ π΄ β dom card) β (π₯ βΊ π΄ β π₯ β (cardβπ΄))) | |
5 | 4 | ancoms 458 | . . . . 5 β’ ((π΄ β dom card β§ π₯ β On) β (π₯ βΊ π΄ β π₯ β (cardβπ΄))) |
6 | 5 | pm5.32da 578 | . . . 4 β’ (π΄ β dom card β ((π₯ β On β§ π₯ βΊ π΄) β (π₯ β On β§ π₯ β (cardβπ΄)))) |
7 | 3, 6 | bitr4id 290 | . . 3 β’ (π΄ β dom card β (π₯ β (cardβπ΄) β (π₯ β On β§ π₯ βΊ π΄))) |
8 | 7 | eqabdv 2861 | . 2 β’ (π΄ β dom card β (cardβπ΄) = {π₯ β£ (π₯ β On β§ π₯ βΊ π΄)}) |
9 | df-rab 3427 | . 2 β’ {π₯ β On β£ π₯ βΊ π΄} = {π₯ β£ (π₯ β On β§ π₯ βΊ π΄)} | |
10 | 8, 9 | eqtr4di 2784 | 1 β’ (π΄ β dom card β (cardβπ΄) = {π₯ β On β£ π₯ βΊ π΄}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 {cab 2703 {crab 3426 class class class wbr 5141 dom cdm 5669 Oncon0 6358 βcfv 6537 βΊ csdm 8940 cardccrd 9932 |
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-pow 5356 ax-pr 5420 ax-un 7722 |
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 6361 df-on 6362 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-card 9936 |
This theorem is referenced by: ondomon 10560 alephsuc3 10577 |
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