Theorem preqsn 4393

Description: Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) (Proof shortened by JJ, 23-Jul-2021.)

Hypotheses

Ref	Expression
preqsn.1	|- A e. _V
preqsn.2	|- B e. _V
preqsn.3	|- C e. _V

Assertion

Ref	Expression
preqsn	|- ( { A , B } = { C } <-> ( A = B /\ B = C ) )

Proof of Theorem preqsn

Step	Hyp	Ref	Expression
1		dfsn2 4190	. . 3 |- { C } = { C , C }
2	1	eqeq2i 2634	. 2 |- ( { A , B } = { C } <-> { A , B } = { C , C } )
3		oridm 536	. . 3 |- ( ( ( A = C /\ B = C ) \/ ( A = C /\ B = C ) ) <-> ( A = C /\ B = C ) )
4		preqsn.1	. . . 4 |- A e. _V
5		preqsn.2	. . . 4 |- B e. _V
6		preqsn.3	. . . 4 |- C e. _V
7	4, 5, 6, 6	preq12b 4382	. . 3 |- ( { A , B } = { C , C } <-> ( ( A = C /\ B = C ) \/ ( A = C /\ B = C ) ) )
8		eqeq2 2633	. . . 4 |- ( B = C -> ( A = B <-> A = C ) )
9	8	pm5.32ri 670	. . 3 |- ( ( A = B /\ B = C ) <-> ( A = C /\ B = C ) )
10	3, 7, 9	3bitr4i 292	. 2 |- ( { A , B } = { C , C } <-> ( A = B /\ B = C ) )
11	2, 10	bitri 264	1 |- ( { A , B } = { C } <-> ( A = B /\ B = C ) )

Syntax hints:   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {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-un 3579  df-sn 4178  df-pr 4180

This theorem is referenced by:  opeqsn  4967  propeqop  4970  propssopi  4971  relop  5272  hash2prde  13252  symg2bas  17818