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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 4636 | . 2 | |
2 | simpr 109 | . 2 | |
3 | vex 2684 | . . . . . 6 | |
4 | vex 2684 | . . . . . 6 | |
5 | 3, 4 | uniopel 4173 | . . . . 5 |
6 | 5 | a1i 9 | . . . 4 |
7 | eleq1 2200 | . . . 4 | |
8 | unieq 3740 | . . . . 5 | |
9 | 8 | eleq1d 2206 | . . . 4 |
10 | 6, 7, 9 | 3imtr4d 202 | . . 3 |
11 | 10 | exlimivv 1868 | . 2 |
12 | 1, 2, 11 | sylc 62 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wex 1468 wcel 1480 cop 3525 cuni 3731 wrel 4539 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-opab 3985 df-xp 4540 df-rel 4541 |
This theorem is referenced by: (None) |
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