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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 4807. (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 488 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V) | |
2 | opprc 4807 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = ∅) | |
3 | 1, 2 | nsyl5 162 | 1 ⊢ (¬ 𝐵 ∈ V → 〈𝐴, 𝐵〉 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3408 ∅c0 4237 〈cop 4547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3410 df-dif 3869 df-nul 4238 df-if 4440 df-op 4548 |
This theorem is referenced by: snopeqop 5389 dmsnopss 6077 strle1 16711 |
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