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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 4812. (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 487 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V) | |
2 | 1 | con3i 157 | . 2 ⊢ (¬ 𝐵 ∈ V → ¬ (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
3 | opprc 4812 | . 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 398 = wceq 1537 ∈ wcel 2114 Vcvv 3486 ∅c0 4279 〈cop 4559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-dif 3927 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-op 4560 |
This theorem is referenced by: snopeqop 5382 dmsnopss 6057 strle1 16575 |
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