Theorem prprc2 4123

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

Assertion

Ref	Expression
prprc2	⊢ (¬ A ∈ V → {B, A} = {B})

Proof of Theorem prprc2

Step	Hyp	Ref	Expression
1		df-pr 3743	. 2 ⊢ {B, A} = ({B} ∪ {A})
2		snprc 3789	. . . . 5 ⊢ (¬ A ∈ V ↔ {A} = ∅)
3	2	biimpi 186	. . . 4 ⊢ (¬ A ∈ V → {A} = ∅)
4	3	uneq2d 3419	. . 3 ⊢ (¬ A ∈ V → ({B} ∪ {A}) = ({B} ∪ ∅))
5		un0 3576	. . 3 ⊢ ({B} ∪ ∅) = {B}
6	4, 5	syl6eq 2401	. 2 ⊢ (¬ A ∈ V → ({B} ∪ {A}) = {B})
7	1, 6	syl5eq 2397	1 ⊢ (¬ A ∈ V → {B, A} = {B})

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∪ cun 3208  ∅c0 3551  {csn 3738  {cpr 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-nul 3552  df-sn 3742  df-pr 3743

This theorem is referenced by:  prprc1  4124