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Theorem brne0 4854
Description: If two sets are in a binary relation, the relation cannot be empty. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
Assertion
Ref Expression
brne0 (𝐴𝑅𝐵𝑅 ≠ ∅)

Proof of Theorem brne0
StepHypRef Expression
1 df-br 4805 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2 ne0i 4064 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝑅𝑅 ≠ ∅)
31, 2sylbi 207 1 (𝐴𝑅𝐵𝑅 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2139  wne 2932  c0 4058  cop 4327   class class class wbr 4804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-v 3342  df-dif 3718  df-nul 4059  df-br 4805
This theorem is referenced by:  epn0  5184  brfvopabrbr  6441  bropfvvvvlem  7424  brfvimex  38826  brovmptimex  38827  clsneibex  38902  neicvgbex  38912
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