![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > brne0 | Structured version Visualization version GIF version |
Description: If two sets are in a binary relation, the relation cannot be empty. (Contributed by Alexander van der Vekens, 7-Jul-2018.) |
Ref | Expression |
---|---|
brne0 | ⊢ (𝐴𝑅𝐵 → 𝑅 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5031 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | ne0i 4250 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ 𝑅 → 𝑅 ≠ ∅) | |
3 | 1, 2 | sylbi 220 | 1 ⊢ (𝐴𝑅𝐵 → 𝑅 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 〈cop 4531 class class class wbr 5030 |
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-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-dif 3884 df-nul 4244 df-br 5031 |
This theorem is referenced by: epn0 5435 brfvopabrbr 6742 bropfvvvvlem 7769 brfvimex 40729 brovmptimex 40730 clsneibex 40805 neicvgbex 40815 |
Copyright terms: Public domain | W3C validator |