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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 10540. This theorem could be used to give a simpler definition of card in place of df-card 9933. 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 9938 | . . . . . 6 β’ (cardβπ΄) β On | |
2 | 1 | oneli 6478 | . . . . 5 β’ (π₯ β (cardβπ΄) β π₯ β On) |
3 | 2 | pm4.71ri 561 | . . . 4 β’ (π₯ β (cardβπ΄) β (π₯ β On β§ π₯ β (cardβπ΄))) |
4 | cardsdomel 9968 | . . . . . 6 β’ ((π₯ β On β§ π΄ β dom card) β (π₯ βΊ π΄ β π₯ β (cardβπ΄))) | |
5 | 4 | ancoms 459 | . . . . 5 β’ ((π΄ β dom card β§ π₯ β On) β (π₯ βΊ π΄ β π₯ β (cardβπ΄))) |
6 | 5 | pm5.32da 579 | . . . 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 289 | . . 3 β’ (π΄ β dom card β (π₯ β (cardβπ΄) β (π₯ β On β§ π₯ βΊ π΄))) |
8 | 7 | eqabdv 2867 | . 2 β’ (π΄ β dom card β (cardβπ΄) = {π₯ β£ (π₯ β On β§ π₯ βΊ π΄)}) |
9 | df-rab 3433 | . 2 β’ {π₯ β On β£ π₯ βΊ π΄} = {π₯ β£ (π₯ β On β§ π₯ βΊ π΄)} | |
10 | 8, 9 | eqtr4di 2790 | 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 396 = wceq 1541 β wcel 2106 {cab 2709 {crab 3432 class class class wbr 5148 dom cdm 5676 Oncon0 6364 βcfv 6543 βΊ csdm 8937 cardccrd 9929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-card 9933 |
This theorem is referenced by: ondomon 10557 alephsuc3 10574 |
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