Theorem ex-br 30782

Description: Example for df-br 5110. 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 5445	. . . 4 ⊢ ⟨3, 9⟩ ∈ V
2	1	prid2 4729	. . 3 ⊢ ⟨3, 9⟩ ∈ {⟨2, 6⟩, ⟨3, 9⟩}
3		id 23	. . 3 ⊢ (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → 𝑅 = {⟨2, 6⟩, ⟨3, 9⟩})
4	2, 3	eleqtrrid 2870	. 2 ⊢ (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → ⟨3, 9⟩ ∈ 𝑅)
5		df-br 5110	. 2 ⊢ (3𝑅9 ↔ ⟨3, 9⟩ ∈ 𝑅)
6	4, 5	sylibr 237	1 ⊢ (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → 3𝑅9)

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  {cpr 4591  ⟨cop 4595   class class class wbr 5109  2c2 12290  3c3 12291  6c6 12294  9c9 12297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-un 3910  df-in 3912  df-ss 3922  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110

This theorem is referenced by: (None)