| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 4896. (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 484 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V) | |
| 2 | opprc 4896 | . 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 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 〈cop 4632 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-dif 3954 df-ss 3968 df-nul 4334 df-if 4526 df-op 4633 |
| This theorem is referenced by: snopeqop 5511 dmsnopss 6234 strle1 17195 |
| Copyright terms: Public domain | W3C validator |