Theorem snopeqop 4969

Description: Equivalence for an ordered pair equal to a singleton of an ordered pair. (Contributed by AV, 18-Sep-2020.)

Hypotheses

Ref	Expression
snopeqop.a	|- A e. _V
snopeqop.b	|- B e. _V
snopeqop.c	|- C e. _V
snopeqop.d	|- D e. _V

Assertion

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

Proof of Theorem snopeqop

Step	Hyp	Ref	Expression
1		snopeqop.a	. . . 4 |- A e. _V
2		snopeqop.b	. . . 4 |- B e. _V
3		snopeqop.c	. . . 4 |- C e. _V
4	1, 2, 3	opeqsn 4967	. . 3 |- ( <. A , B >. = { C } <-> ( A = B /\ C = { A } ) )
5	4	anbi2i 730	. 2 |- ( ( C = D /\ <. A , B >. = { C } ) <-> ( C = D /\ ( A = B /\ C = { A } ) ) )
6		eqcom 2629	. . 3 |- ( { <. A , B >. } = <. C , D >. <-> <. C , D >. = { <. A , B >. } )
7		snopeqop.d	. . . 4 |- D e. _V
8		opex 4932	. . . 4 |- <. A , B >. e. _V
9	3, 7, 8	opeqsn 4967	. . 3 |- ( <. C , D >. = { <. A , B >. } <-> ( C = D /\ <. A , B >. = { C } ) )
10	6, 9	bitri 264	. 2 |- ( { <. A , B >. } = <. C , D >. <-> ( C = D /\ <. A , B >. = { C } ) )
11		3anan12 1051	. 2 |- ( ( A = B /\ C = D /\ C = { A } ) <-> ( C = D /\ ( A = B /\ C = { A } ) ) )
12	5, 10, 11	3bitr4i 292	1 |- ( { <. A , B >. } = <. C , D >. <-> ( A = B /\ C = D /\ C = { A } ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   <.cop 4183

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  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-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184

This theorem is referenced by:  funopsn  6413  funsneqopsn  6417