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Theorem elopk 4129
 Description: Membership in a Kuratowski ordered pair. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
elopk (A B, C⟫ ↔ (A = {B} A = {B, C}))

Proof of Theorem elopk
StepHypRef Expression
1 df-opk 4058 . . 3 B, C⟫ = {{B}, {B, C}}
21eleq2i 2417 . 2 (A B, C⟫ ↔ A {{B}, {B, C}})
3 snex 4111 . . 3 {B} V
4 prex 4112 . . 3 {B, C} V
53, 4elpr2 3752 . 2 (A {{B}, {B, C}} ↔ (A = {B} A = {B, C}))
62, 5bitri 240 1 (A B, C⟫ ↔ (A = {B} A = {B, C}))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∨ wo 357   = wceq 1642   ∈ wcel 1710  {csn 3737  {cpr 3738  ⟪copk 4057 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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  ax-nin 4078  ax-sn 4087 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-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-opk 4058 This theorem is referenced by:  opkth1g  4130
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