Theorem prprc 1839

Description: An unordered pair containing a proper class equals a singleton.

Assertion

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

Proof of Theorem prprc

Step	Hyp	Ref	Expression
1		snprc 1838	. 2 ⊢ (¬ A ∈ V ↔ {A} = ∅)
2		uneq1 1605	. . 3 ⊢ ({A} = ∅ → ({A} ∪ {B}) = (∅ ∪ {B}))
3		df-pr 1812	. . 3 ⊢ {A, B} = ({A} ∪ {B})
4		uncom 1604	. . . 4 ⊢ (∅ ∪ {B}) = ({B} ∪ ∅)
5		un0 1721	. . . 4 ⊢ ({B} ∪ ∅) = {B}
6	4, 5	eqtr2 1120	. . 3 ⊢ {B} = (∅ ∪ {B})
7	2, 3, 6	3eqtr4g 1147	. 2 ⊢ ({A} = ∅ → {A, B} = {B})
8	1, 7	sylbi 174	1 ⊢ (¬ A ∈ V → {A, B} = {B})

Syntax hints:  ¬ wn 1   → wi 2   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∪ cun 1485  ∅c0 1707  {csn 1808  {cpr 1809

This theorem is referenced by:  prex 1892  opprc1 1905  opprc2 1907

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-nul 1708  df-sn 1811  df-pr 1812

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