![]() |
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 5140. 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 5455 | . . . 4 ⊢ ⟨3, 9⟩ ∈ V | |
2 | 1 | prid2 4760 | . . 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 2832 | . 2 ⊢ (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → ⟨3, 9⟩ ∈ 𝑅) |
5 | df-br 5140 | . 2 ⊢ (3𝑅9 ↔ ⟨3, 9⟩ ∈ 𝑅) | |
6 | 4, 5 | sylibr 233 | 1 ⊢ (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → 3𝑅9) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {cpr 4623 ⟨cop 4627 class class class wbr 5139 2c2 12266 3c3 12267 6c6 12270 9c9 12273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |