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Theorem elopg 5477
Description: Characterization of the elements of an ordered pair. Closed form of elop 5478. (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 4876 . 2 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
2 eleq2 2828 . . 3 (⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}} → (𝐶 ∈ ⟨𝐴, 𝐵⟩ ↔ 𝐶 ∈ {{𝐴}, {𝐴, 𝐵}}))
3 snex 5442 . . . 4 {𝐴} ∈ V
4 prex 5443 . . . 4 {𝐴, 𝐵} ∈ V
53, 4elpr2 4657 . . 3 (𝐶 ∈ {{𝐴}, {𝐴, 𝐵}} ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵}))
62, 5bitrdi 287 . 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 206  wa 395  wo 847   = wceq 1537  wcel 2106  {csn 4631  {cpr 4633  cop 4637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638
This theorem is referenced by:  elop  5478  bj-inftyexpidisj  37193
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