![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 9968 | . . 3 β’ (cardβπ΄) β On | |
2 | eleq1 2817 | . . 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 9981 | . . . 4 β’ (π΄ β On β (cardβπ΄) β π΄) | |
5 | eqss 3995 | . . . . 5 β’ ((cardβπ΄) = π΄ β ((cardβπ΄) β π΄ β§ π΄ β (cardβπ΄))) | |
6 | 5 | baibr 536 | . . . 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 3968 | . . . 4 β’ (π΄ β (cardβπ΄) β βπ₯ β π΄ π₯ β (cardβπ΄)) | |
9 | onelon 6394 | . . . . . 6 β’ ((π΄ β On β§ π₯ β π΄) β π₯ β On) | |
10 | onenon 9973 | . . . . . . 7 β’ (π΄ β On β π΄ β dom card) | |
11 | 10 | adantr 480 | . . . . . 6 β’ ((π΄ β On β§ π₯ β π΄) β π΄ β dom card) |
12 | cardsdomel 9998 | . . . . . 6 β’ ((π₯ β On β§ π΄ β dom card) β (π₯ βΊ π΄ β π₯ β (cardβπ΄))) | |
13 | 9, 11, 12 | syl2anc 583 | . . . . 5 β’ ((π΄ β On β§ π₯ β π΄) β (π₯ βΊ π΄ β π₯ β (cardβπ΄))) |
14 | 13 | ralbidva 3172 | . . . 4 β’ (π΄ β On β (βπ₯ β π΄ π₯ βΊ π΄ β βπ₯ β π΄ π₯ β (cardβπ΄))) |
15 | 8, 14 | bitr4id 290 | . . 3 β’ (π΄ β On β (π΄ β (cardβπ΄) β βπ₯ β π΄ π₯ βΊ π΄)) |
16 | 7, 15 | bitr3d 281 | . 2 β’ (π΄ β On β ((cardβπ΄) = π΄ β βπ₯ β π΄ π₯ βΊ π΄)) |
17 | 3, 16 | biadanii 821 | 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 395 = wceq 1534 β wcel 2099 βwral 3058 β wss 3947 class class class wbr 5148 dom cdm 5678 Oncon0 6369 βcfv 6548 βΊ csdm 8963 cardccrd 9959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6372 df-on 6373 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-card 9963 |
This theorem is referenced by: cardprclem 10003 cardmin2 10023 infxpenlem 10037 alephsuc2 10104 cardmin 10588 alephreg 10606 pwcfsdom 10607 winalim2 10720 gchina 10723 inar1 10799 r1tskina 10806 gruina 10842 iscard5 42966 |
Copyright terms: Public domain | W3C validator |