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| Mirrors > Home > ILE Home > Th. List > opeliunxp2f | Unicode version | ||
| Description: Membership in a union of
Cartesian products, using bound-variable
hypothesis for |
| Ref | Expression |
|---|---|
| opeliunxp2f.f |
|
| opeliunxp2f.e |
|
| Ref | Expression |
|---|---|
| opeliunxp2f |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-br 4045 |
. . 3
| |
| 2 | relxp 4784 |
. . . . . 6
| |
| 3 | 2 | rgenw 2561 |
. . . . 5
|
| 4 | reliun 4796 |
. . . . 5
| |
| 5 | 3, 4 | mpbir 146 |
. . . 4
|
| 6 | 5 | brrelex1i 4718 |
. . 3
|
| 7 | 1, 6 | sylbir 135 |
. 2
|
| 8 | elex 2783 |
. . 3
| |
| 9 | 8 | adantr 276 |
. 2
|
| 10 | nfiu1 3957 |
. . . . 5
| |
| 11 | 10 | nfel2 2361 |
. . . 4
|
| 12 | nfv 1551 |
. . . . 5
| |
| 13 | opeliunxp2f.f |
. . . . . 6
| |
| 14 | 13 | nfel2 2361 |
. . . . 5
|
| 15 | 12, 14 | nfan 1588 |
. . . 4
|
| 16 | 11, 15 | nfbi 1612 |
. . 3
|
| 17 | opeq1 3819 |
. . . . 5
| |
| 18 | 17 | eleq1d 2274 |
. . . 4
|
| 19 | eleq1 2268 |
. . . . 5
| |
| 20 | opeliunxp2f.e |
. . . . . 6
| |
| 21 | 20 | eleq2d 2275 |
. . . . 5
|
| 22 | 19, 21 | anbi12d 473 |
. . . 4
|
| 23 | 18, 22 | bibi12d 235 |
. . 3
|
| 24 | opeliunxp 4730 |
. . 3
| |
| 25 | 16, 23, 24 | vtoclg1f 2832 |
. 2
|
| 26 | 7, 9, 25 | pm5.21nii 706 |
1
|
| 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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ral 2489 df-rex 2490 df-v 2774 df-sbc 2999 df-csb 3094 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-iun 3929 df-br 4045 df-opab 4106 df-xp 4681 df-rel 4682 |
| This theorem is referenced by: fisumcom2 11749 fprodcom2fi 11937 |
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