Theorem preqsn 2486

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 2420	. . 3 |- {C} = {C, C}
2	1	eqeq2i 1485	. 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 2483	. 2 |- ({A, B} = {C, C} <-> ((A = C /\ B = C) \/ (A = C /\ B = C)))
7		oridm 243	. . 3 |- (((A = C /\ B = C) \/ (A = C /\ B = C)) <-> (A = C /\ B = C))
8		eqtr3t 1494	. . . . 5 |- ((A = C /\ B = C) -> A = B)
9		pm3.27 323	. . . . 5 |- ((A = C /\ B = C) -> B = C)
10	8, 9	jca 288	. . . 4 |- ((A = C /\ B = C) -> (A = B /\ B = C))
11		eqtrt 1492	. . . . 5 |- ((A = B /\ B = C) -> A = C)
12		pm3.27 323	. . . . 5 |- ((A = B /\ B = C) -> B = C)
13	11, 12	jca 288	. . . 4 |- ((A = B /\ B = C) -> (A = C /\ B = C))
14	10, 13	impbi 157	. . 3 |- ((A = C /\ B = C) <-> (A = B /\ B = C))
15	7, 14	bitr 173	. 2 |- (((A = C /\ B = C) \/ (A = C /\ B = C)) <-> (A = B /\ B = C))
16	2, 6, 15	3bitr 177	1 |- ({A, B} = {C} <-> (A = B /\ B = C))

Syntax hints:   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811  {csn 2409  {cpr 2410

This theorem is referenced by:  opeqsn 2802  relop 3275

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413

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