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Mirrors > Home > MPE Home > Th. List > elopab | Structured version 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 3087 |
. 2
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2 | opex 4667 |
. . . . 5
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3 | eleq1 2526 |
. . . . 5
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4 | 2, 3 | mpbiri 233 |
. . . 4
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5 | 4 | adantr 465 |
. . 3
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6 | 5 | exlimivv 1690 |
. 2
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7 | eqeq1 2458 |
. . . . 5
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8 | 7 | anbi1d 704 |
. . . 4
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9 | 8 | 2exbidv 1683 |
. . 3
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10 | df-opab 4462 |
. . 3
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11 | 9, 10 | elab2g 3215 |
. 2
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12 | 1, 6, 11 | pm5.21nii 353 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4524 ax-nul 4532 ax-pr 4642 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-v 3080 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-nul 3749 df-if 3903 df-sn 3989 df-pr 3991 df-op 3995 df-opab 4462 |
This theorem is referenced by: opelopabsbALT 4709 opelopabsb 4710 opelopabt 4712 opelopabga 4713 opabn0 4730 iunopab 4735 epelg 4744 elxp 4968 elopaba 5063 elcnv 5127 dfmpt3 5644 fmptsng 6012 0neqopab 6243 opabex3d 6668 opabex3 6669 fsplit 6790 isfunc 14897 qqhval2 26579 eulerpartlemgvv 26926 rtrclreclem.trans 27515 pellexlem5 29345 pellex 29347 elopaelxp 30306 clwlkswlks 30594 fmptsnd 30893 opelopab4 31615 dicelval3 35188 |
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