Theorem oprcl 4619

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 4120	. 2 ⊢ (𝐶 ∈ ⟨𝐴, 𝐵⟩ → ¬ ⟨𝐴, 𝐵⟩ = ∅)
2		opprc 4616	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
3	1, 2	nsyl2 145	1 ⊢ (𝐶 ∈ ⟨𝐴, 𝐵⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157  Vcvv 3385  ∅c0 4115  ⟨cop 4374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-dif 3772  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-op 4375

This theorem is referenced by:  opth1  5134  opth  5135  0nelop  5150