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Theorem elopg 4861
 Description: Characterization of the elements of an ordered pair. Closed form of elop 4862. (Contributed by BJ, 22-Jun-2019.) (Avoid depending on this detail.)
Assertion
Ref Expression
elopg ((𝐴𝑉𝐵𝑊) → (𝐶 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵})))

Proof of Theorem elopg
StepHypRef Expression
1 dfopg 4338 . 2 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
2 eleq2 2677 . . 3 (⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}} → (𝐶 ∈ ⟨𝐴, 𝐵⟩ ↔ 𝐶 ∈ {{𝐴}, {𝐴, 𝐵}}))
3 snex 4835 . . . 4 {𝐴} ∈ V
4 prex 4836 . . . 4 {𝐴, 𝐵} ∈ V
53, 4elpr2 4147 . . 3 (𝐶 ∈ {{𝐴}, {𝐴, 𝐵}} ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵}))
62, 5syl6bb 275 . 2 (⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}} → (𝐶 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵})))
71, 6syl 17 1 ((𝐴𝑉𝐵𝑊) → (𝐶 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {csn 4125  {cpr 4127  ⟨cop 4131 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132 This theorem is referenced by:  elop  4862  bj-inftyexpidisj  32274
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