![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > brrelex2 | Structured version Visualization version GIF version |
Description: If two classes are related by a binary relation, then the second class is a set. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
brrelex2 | ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brrelex12 5568 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
2 | 1 | simprd 499 | 1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 Vcvv 3441 class class class wbr 5030 Rel wrel 5524 |
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-ral 3111 df-rex 3112 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 df-opab 5093 df-xp 5525 df-rel 5526 |
This theorem is referenced by: brrelex2i 5573 releldm 5778 relelrn 5779 elrelimasn 5920 funbrfv 6691 relbrtpos 7886 ertr 8287 erth 8321 pslem 17808 opeldifid 30362 eqvreltr 36002 eqvrelth 36006 frege124d 40462 frege133d 40466 climfv 42333 funbrafv2 43803 |
Copyright terms: Public domain | W3C validator |