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Mirrors > Home > MPE Home > Th. List > ex-br | Structured version Visualization version GIF version |
Description: Example for df-br 5031. 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 5321 | . . . 4 ⊢ 〈3, 9〉 ∈ V | |
2 | 1 | prid2 4659 | . . 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 2897 | . 2 ⊢ (𝑅 = {〈2, 6〉, 〈3, 9〉} → 〈3, 9〉 ∈ 𝑅) |
5 | df-br 5031 | . 2 ⊢ (3𝑅9 ↔ 〈3, 9〉 ∈ 𝑅) | |
6 | 4, 5 | sylibr 237 | 1 ⊢ (𝑅 = {〈2, 6〉, 〈3, 9〉} → 3𝑅9) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {cpr 4527 〈cop 4531 class class class wbr 5030 2c2 11680 3c3 11681 6c6 11684 9c9 11687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 |
This theorem is referenced by: (None) |
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