Theorem preqsn 3157

Description: Equivalence for a pair equal to a singleton.

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 3057	. . 3 |- {C} = {C, C}
2	1	eqeq2i 1894	. 2 |- ({A, B} = {C} <-> {A, B} = {C, C})
3		preqsn.1	. . 3 |- A e. _V
4		preqsn.2	. . 3 |- B e. _V
5		preqsn.3	. . 3 |- C e. _V
6	3, 4, 5, 5	preq12b 3154	. 2 |- ({A, B} = {C, C} <-> ((A = C /\ B = C) \/ (A = C /\ B = C)))
7		oridm 262	. . 3 |- (((A = C /\ B = C) \/ (A = C /\ B = C)) <-> (A = C /\ B = C))
8		eqtr3 1907	. . . . 5 |- ((A = C /\ B = C) -> A = B)
9		simpr 350	. . . . 5 |- ((A = C /\ B = C) -> B = C)
10	8, 9	jca 310	. . . 4 |- ((A = C /\ B = C) -> (A = B /\ B = C))
11		eqtr 1904	. . . . 5 |- ((A = B /\ B = C) -> A = C)
12		simpr 350	. . . . 5 |- ((A = B /\ B = C) -> B = C)
13	11, 12	jca 310	. . . 4 |- ((A = B /\ B = C) -> (A = C /\ B = C))
14	10, 13	impbii 174	. . 3 |- ((A = C /\ B = C) <-> (A = B /\ B = C))
15	7, 14	bitri 190	. 2 |- (((A = C /\ B = C) \/ (A = C /\ B = C)) <-> (A = B /\ B = C))
16	2, 6, 15	3bitri 194	1 |- ({A, B} = {C} <-> (A = B /\ B = C))

Syntax hints:   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  {csn 3044  {cpr 3045

This theorem is referenced by:  opeqsn 3549  relop 4113

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-un 2600  df-sn 3049  df-pr 3050

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