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Mirrors > Home > MPE Home > Th. List > opeliunxp2f | Structured version Visualization version Unicode version |
Description: Membership in a union of Cartesian products, using bound-variable hypothesis for instead of distinct variable conditions as in opeliunxp2 5260. (Contributed by AV, 25-Oct-2020.) |
Ref | Expression |
---|---|
opeliunxp2f.f | |
opeliunxp2f.e |
Ref | Expression |
---|---|
opeliunxp2f |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 4654 | . . 3 | |
2 | relxp 5227 | . . . . . 6 | |
3 | 2 | rgenw 2924 | . . . . 5 |
4 | reliun 5239 | . . . . 5 | |
5 | 3, 4 | mpbir 221 | . . . 4 |
6 | 5 | brrelexi 5158 | . . 3 |
7 | 1, 6 | sylbir 225 | . 2 |
8 | elex 3212 | . . 3 | |
9 | 8 | adantr 481 | . 2 |
10 | nfiu1 4550 | . . . . 5 | |
11 | 10 | nfel2 2781 | . . . 4 |
12 | nfv 1843 | . . . . 5 | |
13 | opeliunxp2f.f | . . . . . 6 | |
14 | 13 | nfel2 2781 | . . . . 5 |
15 | 12, 14 | nfan 1828 | . . . 4 |
16 | 11, 15 | nfbi 1833 | . . 3 |
17 | opeq1 4402 | . . . . 5 | |
18 | 17 | eleq1d 2686 | . . . 4 |
19 | eleq1 2689 | . . . . 5 | |
20 | opeliunxp2f.e | . . . . . 6 | |
21 | 20 | eleq2d 2687 | . . . . 5 |
22 | 19, 21 | anbi12d 747 | . . . 4 |
23 | 18, 22 | bibi12d 335 | . . 3 |
24 | opeliunxp 5170 | . . 3 | |
25 | 16, 23, 24 | vtoclg1f 3265 | . 2 |
26 | 7, 9, 25 | pm5.21nii 368 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wnfc 2751 wral 2912 cvv 3200 csn 4177 cop 4183 ciun 4520 class class class wbr 4653 cxp 5112 wrel 5119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-iun 4522 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 |
This theorem is referenced by: mpt2xeldm 7337 fsumcom2 14505 fprodcom2 14714 |
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