Theorem prprc 4245

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 4243	. 2 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
2		snprc 4197	. . 3 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅)
3	2	biimpi 205	. 2 ⊢ (¬ 𝐵 ∈ V → {𝐵} = ∅)
4	1, 3	sylan9eq 2664	1 ⊢ ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  {csn 4125  {cpr 4127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128

This theorem is referenced by:  usgraedgprv  25905