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Mirrors > Home > MPE Home > Th. List > Mathboxes > iscard5 | Structured version Visualization version GIF version |
Description: Two ways to express the property of being a cardinal number. (Contributed by RP, 8-Nov-2023.) |
Ref | Expression |
---|---|
iscard5 | β’ ((cardβπ΄) = π΄ β (π΄ β On β§ βπ₯ β π΄ Β¬ π₯ β π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscard 9912 | . 2 β’ ((cardβπ΄) = π΄ β (π΄ β On β§ βπ₯ β π΄ π₯ βΊ π΄)) | |
2 | sdomnen 8922 | . . . . 5 β’ (π₯ βΊ π΄ β Β¬ π₯ β π΄) | |
3 | onelss 6360 | . . . . . . . 8 β’ (π΄ β On β (π₯ β π΄ β π₯ β π΄)) | |
4 | ssdomg 8941 | . . . . . . . 8 β’ (π΄ β On β (π₯ β π΄ β π₯ βΌ π΄)) | |
5 | 3, 4 | syld 47 | . . . . . . 7 β’ (π΄ β On β (π₯ β π΄ β π₯ βΌ π΄)) |
6 | 5 | imp 408 | . . . . . 6 β’ ((π΄ β On β§ π₯ β π΄) β π₯ βΌ π΄) |
7 | brsdom 8916 | . . . . . . . 8 β’ (π₯ βΊ π΄ β (π₯ βΌ π΄ β§ Β¬ π₯ β π΄)) | |
8 | 7 | biimpri 227 | . . . . . . 7 β’ ((π₯ βΌ π΄ β§ Β¬ π₯ β π΄) β π₯ βΊ π΄) |
9 | 8 | a1i 11 | . . . . . 6 β’ ((π΄ β On β§ π₯ β π΄) β ((π₯ βΌ π΄ β§ Β¬ π₯ β π΄) β π₯ βΊ π΄)) |
10 | 6, 9 | mpand 694 | . . . . 5 β’ ((π΄ β On β§ π₯ β π΄) β (Β¬ π₯ β π΄ β π₯ βΊ π΄)) |
11 | 2, 10 | impbid2 225 | . . . 4 β’ ((π΄ β On β§ π₯ β π΄) β (π₯ βΊ π΄ β Β¬ π₯ β π΄)) |
12 | 11 | ralbidva 3173 | . . 3 β’ (π΄ β On β (βπ₯ β π΄ π₯ βΊ π΄ β βπ₯ β π΄ Β¬ π₯ β π΄)) |
13 | 12 | pm5.32i 576 | . 2 β’ ((π΄ β On β§ βπ₯ β π΄ π₯ βΊ π΄) β (π΄ β On β§ βπ₯ β π΄ Β¬ π₯ β π΄)) |
14 | 1, 13 | bitri 275 | 1 β’ ((cardβπ΄) = π΄ β (π΄ β On β§ βπ₯ β π΄ Β¬ π₯ β π΄)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3065 β wss 3911 class class class wbr 5106 Oncon0 6318 βcfv 6497 β cen 8881 βΌ cdom 8882 βΊ csdm 8883 cardccrd 9872 |
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 2708 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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 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 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-card 9876 |
This theorem is referenced by: elrncard 41816 |
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