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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 5475 | . . . 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 2846 | . 2 ⊢ (𝑅 = {〈2, 6〉, 〈3, 9〉} → 〈3, 9〉 ∈ 𝑅) |
5 | df-br 5149 | . 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 1537 ∈ wcel 2106 {cpr 4633 〈cop 4637 class class class wbr 5148 2c2 12319 3c3 12320 6c6 12323 9c9 12326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 |
This theorem is referenced by: (None) |
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