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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 5667. (Contributed by Scott Fenton, 11-Apr-2012.) |
Ref | Expression |
---|---|
brv | ⊢ 𝐴V𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 5348 | . 2 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
2 | df-br 5059 | . 2 ⊢ (𝐴V𝐵 ↔ 〈𝐴, 𝐵〉 ∈ V) | |
3 | 1, 2 | mpbir 233 | 1 ⊢ 𝐴V𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 Vcvv 3494 〈cop 4566 class class class wbr 5058 |
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: brsset 33345 brtxpsd 33350 dffun10 33370 elfuns 33371 dfint3 33408 brub 33410 brvdif 35516 |
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