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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 2740 |
. . . . . 6
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2 | vex 2740 |
. . . . . 6
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3 | 1, 2 | opth1 4232 |
. . . . 5
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4 | 3 | exlimiv 1598 |
. . . 4
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5 | opeq1 3776 |
. . . . 5
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6 | opeq2 3777 |
. . . . . . 7
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7 | 6 | eqeq1d 2186 |
. . . . . 6
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8 | 7 | spcegv 2825 |
. . . . 5
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9 | 5, 8 | syl5 32 |
. . . 4
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10 | 4, 9 | impbid2 143 |
. . 3
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11 | 1 | eldm2 4820 |
. . . 4
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12 | 1, 2 | opex 4225 |
. . . . . 6
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13 | 12 | elsn 3607 |
. . . . 5
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14 | 13 | exbii 1605 |
. . . 4
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15 | 11, 14 | bitri 184 |
. . 3
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16 | velsn 3608 |
. . 3
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17 | 10, 15, 16 | 3bitr4g 223 |
. 2
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18 | 17 | eqrdv 2175 |
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-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-dm 4632 |
This theorem is referenced by: dmpropg 5096 dmsnop 5097 rnsnopg 5102 elxp4 5111 fnsng 5258 funprg 5261 funtpg 5262 fntpg 5267 ennnfonelemhdmp1 12380 ennnfonelemkh 12383 setsvala 12463 setsresg 12470 setscom 12472 setsslid 12482 strle1g 12533 |
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