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Mirrors > Home > ILE Home > Th. List > unielrel | Unicode version |
Description: The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.) |
Ref | Expression |
---|---|
unielrel |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrel 4701 | . 2 | |
2 | simpr 109 | . 2 | |
3 | vex 2725 | . . . . . 6 | |
4 | vex 2725 | . . . . . 6 | |
5 | 3, 4 | uniopel 4229 | . . . . 5 |
6 | 5 | a1i 9 | . . . 4 |
7 | eleq1 2227 | . . . 4 | |
8 | unieq 3793 | . . . . 5 | |
9 | 8 | eleq1d 2233 | . . . 4 |
10 | 6, 7, 9 | 3imtr4d 202 | . . 3 |
11 | 10 | exlimivv 1883 | . 2 |
12 | 1, 2, 11 | sylc 62 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wex 1479 wcel 2135 cop 3574 cuni 3784 wrel 4604 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-v 2724 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-opab 4039 df-xp 4605 df-rel 4606 |
This theorem is referenced by: (None) |
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