Theorem ex-br 30519

Description: Example for df-br 5087. 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

Step	Hyp	Ref	Expression
1		opex 5412	. . . 4 ⊢ ⟨3, 9⟩ ∈ V
2	1	prid2 4708	. . 3 ⊢ ⟨3, 9⟩ ∈ {⟨2, 6⟩, ⟨3, 9⟩}
3		id 22	. . 3 ⊢ (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → 𝑅 = {⟨2, 6⟩, ⟨3, 9⟩})
4	2, 3	eleqtrrid 2844	. 2 ⊢ (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → ⟨3, 9⟩ ∈ 𝑅)
5		df-br 5087	. 2 ⊢ (3𝑅9 ↔ ⟨3, 9⟩ ∈ 𝑅)
6	4, 5	sylibr 234	1 ⊢ (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → 3𝑅9)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {cpr 4570  ⟨cop 4574   class class class wbr 5086  2c2 12230  3c3 12231  6c6 12234  9c9 12237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-un 3895  df-in 3897  df-ss 3907  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087

This theorem is referenced by: (None)