Theorem prprc 4706

Description: An unordered pair containing two proper classes is the empty set. (Contributed by NM, 22-Mar-2006.)

Assertion

Ref	Expression
prprc	⊢ ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = ∅)

Proof of Theorem prprc

Step	Hyp	Ref	Expression
1		prprc1 4704	. 2 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
2		snprc 4656	. . 3 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅)
3	2	biimpi 215	. 2 ⊢ (¬ 𝐵 ∈ V → {𝐵} = ∅)
4	1, 3	sylan9eq 2795	1 ⊢ ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1538   ∈ wcel 2103  Vcvv 3436  ∅c0 4261  {csn 4564  {cpr 4566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1968  ax-7 2008  ax-8 2105  ax-9 2113  ax-ext 2706

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1541  df-fal 1551  df-ex 1779  df-sb 2065  df-clab 2713  df-cleq 2727  df-clel 2813  df-v 3438  df-dif 3894  df-un 3896  df-nul 4262  df-sn 4565  df-pr 4567

This theorem is referenced by: (None)