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Theorem elrel 5811
Description: A member of a relation is an ordered pair. (Contributed by NM, 17-Sep-2006.)
Assertion
Ref Expression
elrel ((Rel 𝑅𝐴𝑅) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝑅(𝑥,𝑦)

Proof of Theorem elrel
StepHypRef Expression
1 df-rel 5696 . . . 4 (Rel 𝑅𝑅 ⊆ (V × V))
21biimpi 216 . . 3 (Rel 𝑅𝑅 ⊆ (V × V))
32sselda 3995 . 2 ((Rel 𝑅𝐴𝑅) → 𝐴 ∈ (V × V))
4 elvv 5763 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
53, 4sylib 218 1 ((Rel 𝑅𝐴𝑅) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478  wss 3963  cop 4637   × cxp 5687  Rel wrel 5694
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  df-opab 5211  df-xp 5695  df-rel 5696
This theorem is referenced by:  eliunxp  5851  elinxp  6039  unielrel  6296  dfpo2  6318  frxp  8150  frxp2  8168  rntpos  8263  gsum2d2lem  20006  funen1cnv  35081  fundmpss  35748  sscoid  35895  elfuns  35897  eliunxp2  48179
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