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Mirrors > Home > ILE Home > Th. List > fvsng | Unicode version |
Description: The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.) |
Ref | Expression |
---|---|
fvsng |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1 3778 |
. . . . 5
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2 | 1 | sneqd 3605 |
. . . 4
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3 | id 19 |
. . . 4
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4 | 2, 3 | fveq12d 5522 |
. . 3
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5 | 4 | eqeq1d 2186 |
. 2
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6 | opeq2 3779 |
. . . . 5
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7 | 6 | sneqd 3605 |
. . . 4
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8 | 7 | fveq1d 5517 |
. . 3
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9 | id 19 |
. . 3
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10 | 8, 9 | eqeq12d 2192 |
. 2
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11 | vex 2740 |
. . 3
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12 | vex 2740 |
. . 3
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13 | 11, 12 | fvsn 5711 |
. 2
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14 | 5, 10, 13 | vtocl2g 2801 |
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 4121 ax-pow 4174 ax-pr 4209 |
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-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-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 |
This theorem is referenced by: fsnunfv 5717 fvpr1g 5722 fvpr2g 5723 tfr0dm 6322 fseq1p1m1 10091 1fv 10136 sumsnf 11412 prodsnf 11595 setsslid 12507 mgm1 12743 sgrp1 12770 mnd1 12801 mnd1id 12802 grp1 12930 ring1 13189 |
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