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Theorem elrel 5134
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 5035 . . . 4 (Rel 𝑅𝑅 ⊆ (V × V))
21biimpi 204 . . 3 (Rel 𝑅𝑅 ⊆ (V × V))
32sselda 3567 . 2 ((Rel 𝑅𝐴𝑅) → 𝐴 ∈ (V × V))
4 elvv 5090 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
53, 4sylib 206 1 ((Rel 𝑅𝐴𝑅) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wex 1694  wcel 1976  Vcvv 3172  wss 3539  cop 4130   × cxp 5026  Rel wrel 5033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-opab 4638  df-xp 5034  df-rel 5035
This theorem is referenced by:  eliunxp  5169  elres  5342  unielrel  5563  frxp  7151  rntpos  7229  gsum2d2lem  18141  dfpo2  30704  fundmpss  30716  sscoid  30996  elfuns  30998  eliunxp2  41900
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