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Mirrors > Home > ILE Home > Th. List > endisj | Unicode version |
Description: Any two sets are equinumerous to disjoint sets. Exercise 4.39 of [Mendelson] p. 255. (Contributed by NM, 16-Apr-2004.) |
Ref | Expression |
---|---|
endisj.1 |
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endisj.2 |
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Ref | Expression |
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endisj |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | endisj.1 |
. . . 4
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2 | 0ex 3875 |
. . . 4
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3 | 1, 2 | xpsnen 6231 |
. . 3
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4 | endisj.2 |
. . . 4
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5 | 1on 5947 |
. . . . 5
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6 | 5 | elexi 2561 |
. . . 4
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7 | 4, 6 | xpsnen 6231 |
. . 3
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8 | 3, 7 | pm3.2i 257 |
. 2
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9 | xp01disj 5956 |
. 2
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10 | p0ex 3930 |
. . . 4
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11 | 1, 10 | xpex 4396 |
. . 3
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12 | 6 | snex 3928 |
. . . 4
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13 | 4, 12 | xpex 4396 |
. . 3
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14 | breq1 3758 |
. . . . 5
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15 | breq1 3758 |
. . . . 5
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16 | 14, 15 | bi2anan9 538 |
. . . 4
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17 | ineq12 3127 |
. . . . 5
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18 | 17 | eqeq1d 2045 |
. . . 4
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19 | 16, 18 | anbi12d 442 |
. . 3
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20 | 11, 13, 19 | spc2ev 2642 |
. 2
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21 | 8, 9, 20 | mp2an 402 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-id 4021 df-iord 4069 df-on 4071 df-suc 4074 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-1o 5940 df-en 6158 |
This theorem is referenced by: (None) |
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