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Mirrors > Home > MPE Home > Th. List > opprc | Structured version Visualization version GIF version |
Description: Expansion of an ordered pair when either member is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opprc | ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfopif 4866 | . 2 ⊢ ⟨𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) | |
2 | iffalse 4533 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) = ∅) | |
3 | 1, 2 | eqtrid 2780 | 1 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3470 ∅c0 4318 ifcif 4524 {csn 4624 {cpr 4626 ⟨cop 4630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3472 df-dif 3948 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-op 4631 |
This theorem is referenced by: opprc1 4893 opprc2 4894 oprcl 4895 opnz 5469 brabv 5565 opswap 6227 brtpos 8234 bj-brrelex12ALT 36540 elnonrel 43009 |
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