Theorem elopk 4130

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

Step	Hyp	Ref	Expression
1		df-opk 4059	. . 3 ⊢ ⟪B, C⟫ = {{B}, {B, C}}
2	1	eleq2i 2417	. 2 ⊢ (A ∈ ⟪B, C⟫ ↔ A ∈ {{B}, {B, C}})
3		snex 4112	. . 3 ⊢ {B} ∈ V
4		prex 4113	. . 3 ⊢ {B, C} ∈ V
5	3, 4	elpr2 3753	. 2 ⊢ (A ∈ {{B}, {B, C}} ↔ (A = {B} ∨ A = {B, C}))
6	2, 5	bitri 240	1 ⊢ (A ∈ ⟪B, C⟫ ↔ (A = {B} ∨ A = {B, C}))

Syntax hints:   ↔ wb 176   ∨ wo 357   = wceq 1642   ∈ wcel 1710  {csn 3738  {cpr 3739  ⟪copk 4058

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  ax-nin 4079  ax-sn 4088

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 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-opk 4059

This theorem is referenced by:  opkth1g  4131