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Theorem ex-br 30460
Description: Example for df-br 5149. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
ex-br (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → 3𝑅9)

Proof of Theorem ex-br
StepHypRef Expression
1 opex 5475 . . . 4 ⟨3, 9⟩ ∈ V
21prid2 4768 . . 3 ⟨3, 9⟩ ∈ {⟨2, 6⟩, ⟨3, 9⟩}
3 id 22 . . 3 (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → 𝑅 = {⟨2, 6⟩, ⟨3, 9⟩})
42, 3eleqtrrid 2846 . 2 (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → ⟨3, 9⟩ ∈ 𝑅)
5 df-br 5149 . 2 (3𝑅9 ↔ ⟨3, 9⟩ ∈ 𝑅)
64, 5sylibr 234 1 (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → 3𝑅9)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {cpr 4633  cop 4637   class class class wbr 5148  2c2 12319  3c3 12320  6c6 12323  9c9 12326
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-br 5149
This theorem is referenced by: (None)
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