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Theorem elopg 5341
Description: Characterization of the elements of an ordered pair. Closed form of elop 5342. (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 4784 . 2 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
2 eleq2 2904 . . 3 (⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}} → (𝐶 ∈ ⟨𝐴, 𝐵⟩ ↔ 𝐶 ∈ {{𝐴}, {𝐴, 𝐵}}))
3 snex 5315 . . . 4 {𝐴} ∈ V
4 prex 5316 . . . 4 {𝐴, 𝐵} ∈ V
53, 4elpr2 4573 . . 3 (𝐶 ∈ {{𝐴}, {𝐴, 𝐵}} ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵}))
62, 5syl6bb 290 . 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 209  wa 399  wo 844   = wceq 1538  wcel 2115  {csn 4548  {cpr 4550  cop 4554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pr 5313
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555
This theorem is referenced by:  elop  5342  bj-inftyexpidisj  34533
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