Theorem ex-br 30455

Description: Example for df-br 5097. 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 5410	. . . 4 ⊢ ⟨3, 9⟩ ∈ V
2	1	prid2 4718	. . 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 2841	. 2 ⊢ (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → ⟨3, 9⟩ ∈ 𝑅)
5		df-br 5097	. 2 ⊢ (3𝑅9 ↔ ⟨3, 9⟩ ∈ 𝑅)
6	4, 5	sylibr 234	1 ⊢ (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → 3𝑅9)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  {cpr 4580  ⟨cop 4584   class class class wbr 5096  2c2 12198  3c3 12199  6c6 12202  9c9 12205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097

This theorem is referenced by: (None)