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Mirrors > Home > MPE Home > Th. List > opprc1 | Structured version Visualization version GIF version |
Description: Expansion of an ordered pair when the first member is a proper class. See also opprc 4897. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opprc1 | ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V) | |
2 | opprc 4897 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅) | |
3 | 1, 2 | nsyl5 159 | 1 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ∅c0 4323 ⟨cop 4635 |
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 3473 df-dif 3950 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-op 4636 |
This theorem is referenced by: snopeqop 5508 epelg 5583 brprcneu 6887 brprcneuALT 6888 fmlafvel 34995 bj-inftyexpidisj 36689 eu2ndop1stv 46505 |
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