Theorem oprcl 4865

Description: If an ordered pair has an element, then its arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
oprcl	⊢ (𝐶 ∈ ⟨𝐴, 𝐵⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Proof of Theorem oprcl

Step	Hyp	Ref	Expression
1		n0i 4305	. 2 ⊢ (𝐶 ∈ ⟨𝐴, 𝐵⟩ → ¬ ⟨𝐴, 𝐵⟩ = ∅)
2		opprc 4862	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
3	1, 2	nsyl2 141	1 ⊢ (𝐶 ∈ ⟨𝐴, 𝐵⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ∅c0 4298  ⟨cop 4597

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-dif 3919  df-ss 3933  df-nul 4299  df-if 4491  df-op 4598

This theorem is referenced by:  opth1  5437  opth  5438  0nelop  5458