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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 10545, 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 9971 | . . 3 β’ ((π΄ β dom card β§ π΅ β dom card) β ((cardβπ΄) β (cardβπ΅) β π΄ βΌ π΅)) | |
2 | carddom2 9971 | . . . 4 β’ ((π΅ β dom card β§ π΄ β dom card) β ((cardβπ΅) β (cardβπ΄) β π΅ βΌ π΄)) | |
3 | 2 | ancoms 458 | . . 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 3992 | . . 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 9093 | . 2 β’ ((π΄ βΌ π΅ β§ π΅ βΌ π΄) β π΄ β π΅) | |
8 | 4, 6, 7 | 3bitr3g 313 | 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 395 = wceq 1533 β wcel 2098 β wss 3943 class class class wbr 5141 dom cdm 5669 βcfv 6536 β cen 8935 βΌ cdom 8936 cardccrd 9929 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6360 df-on 6361 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-card 9933 |
This theorem is referenced by: cardsdom2 9982 pm54.43lem 9994 sdom2en01 10296 fin23lem22 10321 fin1a2lem9 10402 pwfseqlem4 10656 hashen 14309 |
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