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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 3487 | . 2 β’ (π΄ β dom card β π΄ β V) | |
2 | isnum2 9942 | . . . 4 β’ (π΄ β dom card β βπ₯ β On π₯ β π΄) | |
3 | rabn0 4380 | . . . 4 β’ ({π₯ β On β£ π₯ β π΄} β β β βπ₯ β On π₯ β π΄) | |
4 | intex 5330 | . . . 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 5145 | . . . . 5 β’ (π¦ = π΄ β (π₯ β π¦ β π₯ β π΄)) | |
8 | 7 | rabbidv 3434 | . . . 4 β’ (π¦ = π΄ β {π₯ β On β£ π₯ β π¦} = {π₯ β On β£ π₯ β π΄}) |
9 | 8 | inteqd 4948 | . . 3 β’ (π¦ = π΄ β β© {π₯ β On β£ π₯ β π¦} = β© {π₯ β On β£ π₯ β π΄}) |
10 | df-card 9936 | . . 3 β’ card = (π¦ β V β¦ β© {π₯ β On β£ π₯ β π¦}) | |
11 | 9, 10 | fvmptg 6990 | . 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 583 | 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 2934 βwrex 3064 {crab 3426 Vcvv 3468 β c0 4317 β© cint 4943 class class class wbr 5141 dom cdm 5669 Oncon0 6358 βcfv 6537 β cen 8938 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-pr 5420 |
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-fv 6545 df-en 8942 df-card 9936 |
This theorem is referenced by: cardid2 9950 oncardval 9952 cardidm 9956 cardne 9962 cardval 10543 |
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