New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  preqr1 GIF version

Theorem preqr1 4124
 Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. (Contributed by NM, 18-Oct-1995.)
Hypotheses
Ref Expression
preqr1.1 A V
preqr1.2 B V
Assertion
Ref Expression
preqr1 ({A, C} = {B, C} → A = B)

Proof of Theorem preqr1
StepHypRef Expression
1 preqr1.1 . . . . 5 A V
21prid1 3827 . . . 4 A {A, C}
3 eleq2 2414 . . . 4 ({A, C} = {B, C} → (A {A, C} ↔ A {B, C}))
42, 3mpbii 202 . . 3 ({A, C} = {B, C} → A {B, C})
51elpr 3751 . . 3 (A {B, C} ↔ (A = B A = C))
64, 5sylib 188 . 2 ({A, C} = {B, C} → (A = B A = C))
7 preqr1.2 . . . . 5 B V
87prid1 3827 . . . 4 B {B, C}
9 eleq2 2414 . . . 4 ({A, C} = {B, C} → (B {A, C} ↔ B {B, C}))
108, 9mpbiri 224 . . 3 ({A, C} = {B, C} → B {A, C})
117elpr 3751 . . 3 (B {A, C} ↔ (B = A B = C))
1210, 11sylib 188 . 2 ({A, C} = {B, C} → (B = A B = C))
13 eqcom 2355 . 2 (A = BB = A)
14 eqeq2 2362 . 2 (A = C → (B = AB = C))
156, 12, 13, 14oplem1 930 1 ({A, C} = {B, C} → A = B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 357   = wceq 1642   ∈ wcel 1710  Vcvv 2859  {cpr 3738 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742 This theorem is referenced by:  preqr2  4125
 Copyright terms: Public domain W3C validator