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Mirrors > Home > ILE Home > Th. List > dmsnopg | Unicode version |
Description: The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
dmsnopg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2554 |
. . . . . 6
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2 | vex 2554 |
. . . . . 6
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3 | 1, 2 | opth1 3964 |
. . . . 5
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4 | 3 | exlimiv 1486 |
. . . 4
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5 | opeq1 3540 |
. . . . 5
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6 | opeq2 3541 |
. . . . . . 7
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7 | 6 | eqeq1d 2045 |
. . . . . 6
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8 | 7 | spcegv 2635 |
. . . . 5
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9 | 5, 8 | syl5 28 |
. . . 4
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10 | 4, 9 | impbid2 131 |
. . 3
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11 | 1 | eldm2 4476 |
. . . 4
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12 | 1, 2 | opex 3957 |
. . . . . 6
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13 | 12 | elsnc 3390 |
. . . . 5
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14 | 13 | exbii 1493 |
. . . 4
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15 | 11, 14 | bitri 173 |
. . 3
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16 | elsn 3382 |
. . 3
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17 | 10, 15, 16 | 3bitr4g 212 |
. 2
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18 | 17 | eqrdv 2035 |
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-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-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-dm 4298 |
This theorem is referenced by: dmpropg 4736 dmsnop 4737 rnsnopg 4742 elxp4 4751 fnsng 4890 funprg 4892 funtpg 4893 fntpg 4898 |
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