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Mirrors > Home > ILE Home > Th. List > ovprc1 | GIF version |
Description: The value of an operation when the first argument is a proper class. (Contributed by NM, 16-Jun-2004.) |
Ref | Expression |
---|---|
ovprc1.1 | ⊢ Rel dom 𝐹 |
Ref | Expression |
---|---|
ovprc1 | ⊢ (¬ 𝐴 ∈ V → (𝐴𝐹𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V) | |
2 | 1 | con3i 621 | . 2 ⊢ (¬ 𝐴 ∈ V → ¬ (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
3 | ovprc1.1 | . . 3 ⊢ Rel dom 𝐹 | |
4 | 3 | ovprc 5806 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅) |
5 | 2, 4 | syl 14 | 1 ⊢ (¬ 𝐴 ∈ V → (𝐴𝐹𝐵) = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∅c0 3363 dom cdm 4539 Rel wrel 4544 (class class class)co 5774 |
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-nul 4054 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-dm 4549 df-iota 5088 df-fv 5131 df-ov 5777 |
This theorem is referenced by: (None) |
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