Theorem ex-br 30855

Description: Example for df-br 5112. 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 5447	. . . 4 ⊢ ⟨3, 9⟩ ∈ V
2	1	prid2 4731	. . 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 2872	. 2 ⊢ (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → ⟨3, 9⟩ ∈ 𝑅)
5		df-br 5112	. 2 ⊢ (3𝑅9 ↔ ⟨3, 9⟩ ∈ 𝑅)
6	4, 5	sylibr 237	1 ⊢ (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → 3𝑅9)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  {cpr 4593  ⟨cop 4597   class class class wbr 5111  2c2 12312  3c3 12313  6c6 12316  9c9 12319

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-un 3911  df-in 3913  df-ss 3923  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112

This theorem is used by: (None)