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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 4902. (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 481 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V) | |
2 | opprc 4902 | . 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 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ∅c0 4325 〈cop 4639 |
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 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-dif 3950 df-ss 3964 df-nul 4326 df-if 4534 df-op 4640 |
This theorem is referenced by: snopeqop 5512 epelg 5587 brprcneu 6891 brprcneuALT 6892 fmlafvel 35213 bj-inftyexpidisj 36917 eu2ndop1stv 46738 |
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