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Mirrors > Home > MPE Home > Th. List > oprcl | Structured version Visualization version GIF version |
Description: If an ordered pair has an element, then its arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
oprcl | ⊢ (𝐶 ∈ 〈𝐴, 𝐵〉 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 4298 | . 2 ⊢ (𝐶 ∈ 〈𝐴, 𝐵〉 → ¬ 〈𝐴, 𝐵〉 = ∅) | |
2 | opprc 4819 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = ∅) | |
3 | 1, 2 | nsyl2 143 | 1 ⊢ (𝐶 ∈ 〈𝐴, 𝐵〉 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 〈cop 4566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-dif 3938 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-op 4567 |
This theorem is referenced by: opth1 5359 opth 5360 0nelop 5378 |
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