|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > ex-br | Structured version Visualization version GIF version | ||
| Description: Example for df-br 5144. 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 5469 | . . . 4 ⊢ 〈3, 9〉 ∈ V | |
| 2 | 1 | prid2 4763 | . . 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 2848 | . 2 ⊢ (𝑅 = {〈2, 6〉, 〈3, 9〉} → 〈3, 9〉 ∈ 𝑅) | 
| 5 | df-br 5144 | . 2 ⊢ (3𝑅9 ↔ 〈3, 9〉 ∈ 𝑅) | |
| 6 | 4, 5 | sylibr 234 | 1 ⊢ (𝑅 = {〈2, 6〉, 〈3, 9〉} → 3𝑅9) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {cpr 4628 〈cop 4632 class class class wbr 5143 2c2 12321 3c3 12322 6c6 12325 9c9 12328 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |