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Mirrors > Home > MPE Home > Th. List > opprc2 | Structured version Visualization version GIF version |
Description: Expansion of an ordered pair when the second member is a proper class. See also opprc 4857. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opprc2 | ⊢ (¬ 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 486 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V) | |
2 | opprc 4857 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅) | |
3 | 1, 2 | nsyl5 159 | 1 ⊢ (¬ 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3447 ∅c0 4286 ⟨cop 4596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3449 df-dif 3917 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-op 4597 |
This theorem is referenced by: snopeqop 5467 dmsnopss 6170 strle1 17038 |
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