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Mirrors > Home > MPE Home > Th. List > elopab | Structured version Visualization 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 3040 |
. 2
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2 | opex 4664 |
. . . . 5
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3 | eleq1 2537 |
. . . . 5
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4 | 2, 3 | mpbiri 241 |
. . . 4
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5 | 4 | adantr 472 |
. . 3
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6 | 5 | exlimivv 1786 |
. 2
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7 | eqeq1 2475 |
. . . . 5
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8 | 7 | anbi1d 719 |
. . . 4
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9 | 8 | 2exbidv 1778 |
. . 3
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10 | df-opab 4455 |
. . 3
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11 | 9, 10 | elab2g 3175 |
. 2
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12 | 1, 6, 11 | pm5.21nii 360 |
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 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pr 4639 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-v 3033 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-sn 3960 df-pr 3962 df-op 3966 df-opab 4455 |
This theorem is referenced by: opelopabsbALT 4710 opelopabsb 4711 opelopabt 4713 opelopabga 4714 opabn0 4732 iunopab 4737 elopabr 4738 epelg 4751 elxp 4856 elopaelxp 4912 elopaba 4952 elcnv 5016 dfmpt3 5708 fmptsng 6101 fmptsnd 6102 0neqopab 6353 opabex3d 6790 opabex3 6791 fsplit 6920 rtrclreclem3 13200 isfunc 15847 usgraop 25156 clwlkswlks 25565 brabgaf 28292 qqhval2 28860 eulerpartlemgvv 29282 poimirlem26 32030 dicelval3 34819 pellexlem5 35748 pellex 35750 opelopab4 36988 griedg0ssusgr 39501 rgrusgrprc 39793 |
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