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 5058 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | ne0i 4297 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ 𝑅 → 𝑅 ≠ ∅) | |
3 | 1, 2 | sylbi 218 | 1 ⊢ (𝐴𝑅𝐵 → 𝑅 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ≠ wne 3013 ∅c0 4288 〈cop 4563 class class class wbr 5057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-ne 3014 df-dif 3936 df-nul 4289 df-br 5058 |
This theorem is referenced by: epn0 5464 brfvopabrbr 6758 bropfvvvvlem 7775 brfvimex 40254 brovmptimex 40255 clsneibex 40330 neicvgbex 40340 |
Copyright terms: Public domain | W3C validator |