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Theorem elcnv 5467
Description: Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 24-Mar-1998.)
Assertion
Ref Expression
elcnv (𝐴𝑅 ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑦𝑅𝑥))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦

Proof of Theorem elcnv
StepHypRef Expression
1 df-cnv 5285 . . 3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥}
21eleq2i 2836 . 2 (𝐴𝑅𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥})
3 elopab 5144 . 2 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥} ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑦𝑅𝑥))
42, 3bitri 266 1 (𝐴𝑅 ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑦𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  cop 4340   class class class wbr 4809  {copab 4871  ccnv 5276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-opab 4872  df-cnv 5285
This theorem is referenced by:  elcnv2  5468  gsummpt2co  30227
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