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Mirrors > Home > ILE Home > Th. List > carden2bex | Unicode version |
Description: If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.) |
Ref | Expression |
---|---|
carden2bex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enen2 6817 | . . . . 5 | |
2 | 1 | rabbidv 2719 | . . . 4 |
3 | 2 | inteqd 3834 | . . 3 |
4 | 3 | adantr 274 | . 2 |
5 | cardval3ex 7155 | . . 3 | |
6 | 5 | adantl 275 | . 2 |
7 | entr 6760 | . . . . . 6 | |
8 | 7 | expcom 115 | . . . . 5 |
9 | 8 | reximdv 2571 | . . . 4 |
10 | 9 | imp 123 | . . 3 |
11 | cardval3ex 7155 | . . 3 | |
12 | 10, 11 | syl 14 | . 2 |
13 | 4, 6, 12 | 3eqtr4d 2213 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wrex 2449 crab 2452 cint 3829 class class class wbr 3987 con0 4346 cfv 5196 cen 6714 ccrd 7149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-er 6511 df-en 6717 df-card 7150 |
This theorem is referenced by: (None) |
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