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Mirrors > Home > ILE Home > Th. List > elopab | Unicode version |
Description: Membership in a class abstraction of pairs. (Contributed by NM, 24-Mar-1998.) |
Ref | Expression |
---|---|
elopab |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2700 |
. 2
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2 | vex 2692 |
. . . . . 6
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3 | vex 2692 |
. . . . . 6
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4 | 2, 3 | opex 4159 |
. . . . 5
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5 | eleq1 2203 |
. . . . 5
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6 | 4, 5 | mpbiri 167 |
. . . 4
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7 | 6 | adantr 274 |
. . 3
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8 | 7 | exlimivv 1869 |
. 2
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9 | eqeq1 2147 |
. . . . 5
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10 | 9 | anbi1d 461 |
. . . 4
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11 | 10 | 2exbidv 1841 |
. . 3
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12 | df-opab 3998 |
. . 3
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13 | 11, 12 | elab2g 2835 |
. 2
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14 | 1, 8, 13 | pm5.21nii 694 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-opab 3998 |
This theorem is referenced by: opelopabsbALT 4189 opelopabsb 4190 opelopabt 4192 opelopabga 4193 opabm 4210 iunopab 4211 epelg 4220 elxp 4564 elco 4713 elcnv 4724 dfmpt3 5253 0neqopab 5824 brabvv 5825 opabex3d 6027 opabex3 6028 |
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