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Theorem elop 4148
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 3699 . . 3 𝐵, 𝐶⟩ = {{𝐵}, {𝐵, 𝐶}}
43eleq2i 2204 . 2 (𝐴 ∈ ⟨𝐵, 𝐶⟩ ↔ 𝐴 ∈ {{𝐵}, {𝐵, 𝐶}})
5 elop.1 . . 3 𝐴 ∈ V
65elpr 3543 . 2 (𝐴 ∈ {{𝐵}, {𝐵, 𝐶}} ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶}))
74, 6bitri 183 1 (𝐴 ∈ ⟨𝐵, 𝐶⟩ ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶}))
Colors of variables: wff set class
Syntax hints:  wb 104  wo 697   = wceq 1331  wcel 1480  Vcvv 2681  {csn 3522  {cpr 3523  cop 3525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531
This theorem is referenced by:  relop  4684  bdop  13062
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