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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 5149 | . 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅) | |
2 | ne0i 4335 | . 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝑅 → 𝑅 ≠ ∅) | |
3 | 1, 2 | sylbi 216 | 1 ⊢ (𝐴𝑅𝐵 → 𝑅 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ≠ wne 2937 ∅c0 4323 ⟨cop 4635 class class class wbr 5148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-dif 3950 df-nul 4324 df-br 5149 |
This theorem is referenced by: epn0 5587 brfvopabrbr 7002 bropfvvvvlem 8096 brfvimex 43456 brovmptimex 43457 clsneibex 43532 neicvgbex 43542 |
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