![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4870 | . 2 ⊢ ⟨𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) | |
2 | iffalse 4537 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) = ∅) | |
3 | 1, 2 | eqtrid 2784 | 1 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∅c0 4322 ifcif 4528 {csn 4628 {cpr 4630 ⟨cop 4634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-dif 3951 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-op 4635 |
This theorem is referenced by: opprc1 4897 opprc2 4898 oprcl 4899 opnz 5473 brabv 5569 opswap 6228 brtpos 8219 bj-brrelex12ALT 35943 elnonrel 42326 |
Copyright terms: Public domain | W3C validator |