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Mirrors > Home > MPE Home > Th. List > carden2 | Structured version Visualization version GIF version |
Description: Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 10582, the Axiom of Choice is not required. (Contributed by Mario Carneiro, 22-Sep-2013.) |
Ref | Expression |
---|---|
carden2 | β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΄) = (cardβπ΅) β π΄ β π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | carddom2 10008 | . . 3 β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΄) β (cardβπ΅) β π΄ βΌ π΅)) | |
2 | carddom2 10008 | . . . 4 β’ ((π΅ β dom card β§ π΄ β dom card) β ((cardβπ΅) β (cardβπ΄) β π΅ βΌ π΄)) | |
3 | 2 | ancoms 457 | . . 3 β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΅) β (cardβπ΄) β π΅ βΌ π΄)) |
4 | 1, 3 | anbi12d 630 | . 2 β’ ((π΄ β dom card β§ π΅ β dom card) β (((cardβπ΄) β (cardβπ΅) β§ (cardβπ΅) β (cardβπ΄)) β (π΄ βΌ π΅ β§ π΅ βΌ π΄))) |
5 | eqss 3997 | . . 3 β’ ((cardβπ΄) = (cardβπ΅) β ((cardβπ΄) β (cardβπ΅) β§ (cardβπ΅) β (cardβπ΄))) | |
6 | 5 | bicomi 223 | . 2 β’ (((cardβπ΄) β (cardβπ΅) β§ (cardβπ΅) β (cardβπ΄)) β (cardβπ΄) = (cardβπ΅)) |
7 | sbthb 9125 | . 2 β’ ((π΄ βΌ π΅ β§ π΅ βΌ π΄) β π΄ β π΅) | |
8 | 4, 6, 7 | 3bitr3g 312 | 1 β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΄) = (cardβπ΅) β π΄ β π΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 β wss 3949 class class class wbr 5152 dom cdm 5682 βcfv 6553 β cen 8967 βΌ cdom 8968 cardccrd 9966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-card 9970 |
This theorem is referenced by: cardsdom2 10019 pm54.43lem 10031 sdom2en01 10333 fin23lem22 10358 fin1a2lem9 10439 pwfseqlem4 10693 hashen 14346 |
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