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Theorem elop 4246
Description: An ordered pair has two elements. Exercise 3 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
elop.1 𝐴 ∈ V
elop.2 𝐵 ∈ V
elop.3 𝐶 ∈ V
Assertion
Ref Expression
elop (𝐴 ∈ ⟨𝐵, 𝐶⟩ ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶}))

Proof of Theorem elop
StepHypRef Expression
1 elop.2 . . . 4 𝐵 ∈ V
2 elop.3 . . . 4 𝐶 ∈ V
31, 2dfop 3792 . . 3 𝐵, 𝐶⟩ = {{𝐵}, {𝐵, 𝐶}}
43eleq2i 2256 . 2 (𝐴 ∈ ⟨𝐵, 𝐶⟩ ↔ 𝐴 ∈ {{𝐵}, {𝐵, 𝐶}})
5 elop.1 . . 3 𝐴 ∈ V
65elpr 3628 . 2 (𝐴 ∈ {{𝐵}, {𝐵, 𝐶}} ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶}))
74, 6bitri 184 1 (𝐴 ∈ ⟨𝐵, 𝐶⟩ ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶}))
Colors of variables: wff set class
Syntax hints:  wb 105  wo 709   = wceq 1364  wcel 2160  Vcvv 2752  {csn 3607  {cpr 3608  cop 3610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616
This theorem is referenced by:  relop  4792  bdop  15064
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