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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 4424. (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 473 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V) | |
2 | 1 | con3i 150 | . 2 ⊢ (¬ 𝐴 ∈ V → ¬ (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
3 | opprc 4424 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = ∅) | |
4 | 2, 3 | syl 17 | 1 ⊢ (¬ 𝐴 ∈ V → 〈𝐴, 𝐵〉 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 〈cop 4183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-op 4184 |
This theorem is referenced by: brprcneu 6184 bj-inftyexpidisj 33097 eu2ndop1stv 41202 |
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