![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 109 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V) | |
2 | 1 | con3i 633 | . 2 ⊢ (¬ 𝐴 ∈ V → ¬ (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
3 | ovprc1.1 | . . 3 ⊢ Rel dom 𝐹 | |
4 | 3 | ovprc 5931 | . 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 104 = wceq 1364 ∈ wcel 2160 Vcvv 2752 ∅c0 3437 dom cdm 4644 Rel wrel 4649 (class class class)co 5896 |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-xp 4650 df-rel 4651 df-dm 4654 df-iota 5196 df-fv 5243 df-ov 5899 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |