![]() |
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 9885 | . . 3 β’ (cardβπ΄) β On | |
2 | eleq1 2822 | . . 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 9898 | . . . 4 β’ (π΄ β On β (cardβπ΄) β π΄) | |
5 | eqss 3960 | . . . . 5 β’ ((cardβπ΄) = π΄ β ((cardβπ΄) β π΄ β§ π΄ β (cardβπ΄))) | |
6 | 5 | baibr 538 | . . . 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 3933 | . . . 4 β’ (π΄ β (cardβπ΄) β βπ₯ β π΄ π₯ β (cardβπ΄)) | |
9 | onelon 6343 | . . . . . 6 β’ ((π΄ β On β§ π₯ β π΄) β π₯ β On) | |
10 | onenon 9890 | . . . . . . 7 β’ (π΄ β On β π΄ β dom card) | |
11 | 10 | adantr 482 | . . . . . 6 β’ ((π΄ β On β§ π₯ β π΄) β π΄ β dom card) |
12 | cardsdomel 9915 | . . . . . 6 β’ ((π₯ β On β§ π΄ β dom card) β (π₯ βΊ π΄ β π₯ β (cardβπ΄))) | |
13 | 9, 11, 12 | syl2anc 585 | . . . . 5 β’ ((π΄ β On β§ π₯ β π΄) β (π₯ βΊ π΄ β π₯ β (cardβπ΄))) |
14 | 13 | ralbidva 3169 | . . . 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 397 = wceq 1542 β wcel 2107 βwral 3061 β wss 3911 class class class wbr 5106 dom cdm 5634 Oncon0 6318 βcfv 6497 βΊ csdm 8885 cardccrd 9876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-card 9880 |
This theorem is referenced by: cardprclem 9920 cardmin2 9940 infxpenlem 9954 alephsuc2 10021 cardmin 10505 alephreg 10523 pwcfsdom 10524 winalim2 10637 gchina 10640 inar1 10716 r1tskina 10723 gruina 10759 iscard5 41896 |
Copyright terms: Public domain | W3C validator |