Theorem preqr1 1872

Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal.

Hypotheses

Ref	Expression
preqr1.1	|- A e. V
preqr1.2	|- B e. V

Assertion

Ref	Expression
preqr1	|- ({A, C} = {B, C} -> A = B)

Proof of Theorem preqr1

Step	Hyp	Ref	Expression
1		preqr1.1	. . . . 5 |- A e. V
2	1	pri1 1841	. . . 4 |- A e. {A, C}
3		eleq2 1150	. . . 4 |- ({A, C} = {B, C} -> (A e. {A, C} <-> A e. {B, C}))
4	2, 3	mpbii 168	. . 3 |- ({A, C} = {B, C} -> A e. {B, C})
5	1	elpr 1823	. . 3 |- (A e. {B, C} <-> (A = B \/ A = C))
6	4, 5	sylib 173	. 2 |- ({A, C} = {B, C} -> (A = B \/ A = C))
7		preqr1.2	. . . . 5 |- B e. V
8	7	pri1 1841	. . . 4 |- B e. {B, C}
9		eleq2 1150	. . . 4 |- ({A, C} = {B, C} -> (B e. {A, C} <-> B e. {B, C}))
10	8, 9	mpbiri 169	. . 3 |- ({A, C} = {B, C} -> B e. {A, C})
11	7	elpr 1823	. . 3 |- (B e. {A, C} <-> (B = A \/ B = C))
12	10, 11	sylib 173	. 2 |- ({A, C} = {B, C} -> (B = A \/ B = C))
13		cleqcom 1103	. 2 |- (A = B <-> B = A)
14		cleq2 1110	. 2 |- (A = C -> (B = A <-> B = C))
15	6, 12, 13, 14	oplem1 578	1 |- ({A, C} = {B, C} -> A = B)

Syntax hints:   -> wi 2   \/ wo 195   = wceq 1091   e. wcel 1092  Vcvv 1348  {cpr 1809

This theorem is referenced by:  prer2 1873  opthwiener 1914

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-un 1490  df-sn 1811  df-pr 1812

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