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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enen2 6837 |
. . . . 5
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2 | 1 | rabbidv 2726 |
. . . 4
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3 | 2 | inteqd 3849 |
. . 3
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4 | 3 | adantr 276 |
. 2
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5 | cardval3ex 7180 |
. . 3
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6 | 5 | adantl 277 |
. 2
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7 | entr 6780 |
. . . . . 6
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8 | 7 | expcom 116 |
. . . . 5
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9 | 8 | reximdv 2578 |
. . . 4
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10 | 9 | imp 124 |
. . 3
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11 | cardval3ex 7180 |
. . 3
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12 | 10, 11 | syl 14 |
. 2
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13 | 4, 6, 12 | 3eqtr4d 2220 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5176 df-fun 5216 df-fn 5217 df-f 5218 df-f1 5219 df-fo 5220 df-f1o 5221 df-fv 5222 df-er 6531 df-en 6737 df-card 7175 |
This theorem is referenced by: (None) |
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