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Mirrors > Home > MPE Home > Th. List > ex-br | Structured version Visualization version GIF version |
Description: Example for df-br 5149. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
ex-br | ⊢ (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → 3𝑅9) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 5466 | . . . 4 ⊢ ⟨3, 9⟩ ∈ V | |
2 | 1 | prid2 4768 | . . 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 2836 | . 2 ⊢ (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → ⟨3, 9⟩ ∈ 𝑅) |
5 | df-br 5149 | . 2 ⊢ (3𝑅9 ↔ ⟨3, 9⟩ ∈ 𝑅) | |
6 | 4, 5 | sylibr 233 | 1 ⊢ (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → 3𝑅9) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 {cpr 4631 ⟨cop 4635 class class class wbr 5148 2c2 12297 3c3 12298 6c6 12301 9c9 12304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 |
This theorem is referenced by: (None) |
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