![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > brv | Structured version Visualization version GIF version |
Description: Two classes are always in relation by V. This is simply equivalent to 〈𝐴, 𝐵〉 ∈ V, and does not imply that V is a relation: see nrelv 5457. (Contributed by Scott Fenton, 11-Apr-2012.) |
Ref | Expression |
---|---|
brv | ⊢ 𝐴V𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 5152 | . 2 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
2 | df-br 4873 | . 2 ⊢ (𝐴V𝐵 ↔ 〈𝐴, 𝐵〉 ∈ V) | |
3 | 1, 2 | mpbir 223 | 1 ⊢ 𝐴V𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2166 Vcvv 3413 〈cop 4402 class class class wbr 4872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pr 5126 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-v 3415 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-sn 4397 df-pr 4399 df-op 4403 df-br 4873 |
This theorem is referenced by: brsset 32534 brtxpsd 32539 dffun10 32559 elfuns 32560 dfint3 32597 brub 32599 brvdif 34578 |
Copyright terms: Public domain | W3C validator |