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Mirrors > Home > MPE Home > Th. List > iscard | Structured version Visualization version GIF version |
Description: Two ways to express the property of being a cardinal number. (Contributed by Mario Carneiro, 15-Jan-2013.) |
Ref | Expression |
---|---|
iscard | β’ ((cardβπ΄) = π΄ β (π΄ β On β§ βπ₯ β π΄ π₯ βΊ π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardon 9935 | . . 3 β’ (cardβπ΄) β On | |
2 | eleq1 2821 | . . 3 β’ ((cardβπ΄) = π΄ β ((cardβπ΄) β On β π΄ β On)) | |
3 | 1, 2 | mpbii 232 | . 2 β’ ((cardβπ΄) = π΄ β π΄ β On) |
4 | cardonle 9948 | . . . 4 β’ (π΄ β On β (cardβπ΄) β π΄) | |
5 | eqss 3996 | . . . . 5 β’ ((cardβπ΄) = π΄ β ((cardβπ΄) β π΄ β§ π΄ β (cardβπ΄))) | |
6 | 5 | baibr 537 | . . . 4 β’ ((cardβπ΄) β π΄ β (π΄ β (cardβπ΄) β (cardβπ΄) = π΄)) |
7 | 4, 6 | syl 17 | . . 3 β’ (π΄ β On β (π΄ β (cardβπ΄) β (cardβπ΄) = π΄)) |
8 | dfss3 3969 | . . . 4 β’ (π΄ β (cardβπ΄) β βπ₯ β π΄ π₯ β (cardβπ΄)) | |
9 | onelon 6386 | . . . . . 6 β’ ((π΄ β On β§ π₯ β π΄) β π₯ β On) | |
10 | onenon 9940 | . . . . . . 7 β’ (π΄ β On β π΄ β dom card) | |
11 | 10 | adantr 481 | . . . . . 6 β’ ((π΄ β On β§ π₯ β π΄) β π΄ β dom card) |
12 | cardsdomel 9965 | . . . . . 6 β’ ((π₯ β On β§ π΄ β dom card) β (π₯ βΊ π΄ β π₯ β (cardβπ΄))) | |
13 | 9, 11, 12 | syl2anc 584 | . . . . 5 β’ ((π΄ β On β§ π₯ β π΄) β (π₯ βΊ π΄ β π₯ β (cardβπ΄))) |
14 | 13 | ralbidva 3175 | . . . 4 β’ (π΄ β On β (βπ₯ β π΄ π₯ βΊ π΄ β βπ₯ β π΄ π₯ β (cardβπ΄))) |
15 | 8, 14 | bitr4id 289 | . . 3 β’ (π΄ β On β (π΄ β (cardβπ΄) β βπ₯ β π΄ π₯ βΊ π΄)) |
16 | 7, 15 | bitr3d 280 | . 2 β’ (π΄ β On β ((cardβπ΄) = π΄ β βπ₯ β π΄ π₯ βΊ π΄)) |
17 | 3, 16 | biadanii 820 | 1 β’ ((cardβπ΄) = π΄ β (π΄ β On β§ βπ₯ β π΄ π₯ βΊ π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 396 = wceq 1541 β wcel 2106 βwral 3061 β wss 3947 class class class wbr 5147 dom cdm 5675 Oncon0 6361 βcfv 6540 βΊ csdm 8934 cardccrd 9926 |
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 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
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 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6364 df-on 6365 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-card 9930 |
This theorem is referenced by: cardprclem 9970 cardmin2 9990 infxpenlem 10004 alephsuc2 10071 cardmin 10555 alephreg 10573 pwcfsdom 10574 winalim2 10687 gchina 10690 inar1 10766 r1tskina 10773 gruina 10809 iscard5 42272 |
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