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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 4564. (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 479 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V) | |
2 | 1 | con3i 150 | . 2 ⊢ (¬ 𝐵 ∈ V → ¬ (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
3 | opprc 4564 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = ∅) | |
4 | 2, 3 | syl 17 | 1 ⊢ (¬ 𝐵 ∈ V → 〈𝐴, 𝐵〉 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1620 ∈ wcel 2127 Vcvv 3328 ∅c0 4046 〈cop 4315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-v 3330 df-dif 3706 df-in 3710 df-ss 3717 df-nul 4047 df-if 4219 df-op 4316 |
This theorem is referenced by: dmsnopss 5754 strle1 16146 |
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