Theorem prprc1 4123

Description: An unordered pair of a proper class. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
prprc1	⊢ (¬ A ∈ V → {A, B} = {B})

Proof of Theorem prprc1

Step	Hyp	Ref	Expression
1		prcom 3798	. 2 ⊢ {A, B} = {B, A}
2		prprc2 4122	. 2 ⊢ (¬ A ∈ V → {B, A} = {B})
3	1, 2	syl5eq 2397	1 ⊢ (¬ A ∈ V → {A, B} = {B})

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642   ∈ wcel 1710  Vcvv 2859  {csn 3737  {cpr 3738

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-nul 3551  df-sn 3741  df-pr 3742

This theorem is referenced by: (None)