Theorem ex-br 30508

Description: Example for df-br 5099. 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 4720	. . 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 2843	. 2 ⊢ (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → ⟨3, 9⟩ ∈ 𝑅)
5		df-br 5099	. 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 4582  ⟨cop 4586   class class class wbr 5098  2c2 12202  3c3 12203  6c6 12206  9c9 12209

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 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099

This theorem is referenced by: (None)