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Mirrors > Home > MPE Home > Th. List > ex-br | Structured version Visualization version GIF version |
Description: Example for df-br 5059. 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 5348 | . . . 4 ⊢ 〈3, 9〉 ∈ V | |
2 | 1 | prid2 4692 | . . 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 2920 | . 2 ⊢ (𝑅 = {〈2, 6〉, 〈3, 9〉} → 〈3, 9〉 ∈ 𝑅) |
5 | df-br 5059 | . 2 ⊢ (3𝑅9 ↔ 〈3, 9〉 ∈ 𝑅) | |
6 | 4, 5 | sylibr 236 | 1 ⊢ (𝑅 = {〈2, 6〉, 〈3, 9〉} → 3𝑅9) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {cpr 4562 〈cop 4566 class class class wbr 5058 2c2 11686 3c3 11687 6c6 11690 9c9 11693 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 |
This theorem is referenced by: (None) |
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