Theorem prprc 4302

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 4300	. 2 ⊢ (¬ 𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})
2		snprc 4253	. . 3 ⊢ (¬ 𝐵 ∈ V ↔ {𝐵} = ∅)
3	2	biimpi 206	. 2 ⊢ (¬ 𝐵 ∈ V → {𝐵} = ∅)
4	1, 3	sylan9eq 2676	1 ⊢ ((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ∅c0 3915  {csn 4177  {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180

This theorem is referenced by: (None)