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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 5067 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | ne0i 4300 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ 𝑅 → 𝑅 ≠ ∅) | |
3 | 1, 2 | sylbi 219 | 1 ⊢ (𝐴𝑅𝐵 → 𝑅 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ≠ wne 3016 ∅c0 4291 〈cop 4573 class class class wbr 5066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-ne 3017 df-dif 3939 df-nul 4292 df-br 5067 |
This theorem is referenced by: epn0 5471 brfvopabrbr 6765 bropfvvvvlem 7786 brfvimex 40396 brovmptimex 40397 clsneibex 40472 neicvgbex 40482 |
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