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Mirrors > Home > ILE Home > Th. List > nprrel | GIF version |
Description: No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.) |
Ref | Expression |
---|---|
nprrel.1 | ⊢ Rel 𝑅 |
nprrel.2 | ⊢ ¬ 𝐴 ∈ V |
Ref | Expression |
---|---|
nprrel | ⊢ ¬ 𝐴𝑅𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nprrel.2 | . 2 ⊢ ¬ 𝐴 ∈ V | |
2 | nprrel.1 | . . 3 ⊢ Rel 𝑅 | |
3 | 2 | brrelex1i 4582 | . 2 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ V) |
4 | 1, 3 | mto 651 | 1 ⊢ ¬ 𝐴𝑅𝐵 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 1480 Vcvv 2686 class class class wbr 3929 Rel wrel 4544 |
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-in1 603 ax-in2 604 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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 |
This theorem is referenced by: (None) |
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