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Mirrors > Home > MPE Home > Th. List > ex-br | Structured version Visualization version GIF version |
Description: Example for df-br 5069. 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 5358 | . . . 4 ⊢ 〈3, 9〉 ∈ V | |
2 | 1 | prid2 4701 | . . 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 2922 | . 2 ⊢ (𝑅 = {〈2, 6〉, 〈3, 9〉} → 〈3, 9〉 ∈ 𝑅) |
5 | df-br 5069 | . 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 1537 ∈ wcel 2114 {cpr 4571 〈cop 4575 class class class wbr 5068 2c2 11695 3c3 11696 6c6 11699 9c9 11702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 |
This theorem is referenced by: (None) |
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