MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  relrpss Structured version   Visualization version   GIF version

Theorem relrpss 7085
Description: The proper subset relation is a relation. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Assertion
Ref Expression
relrpss Rel []

Proof of Theorem relrpss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rpss 7084 . 2 [] = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
21relopabi 5384 1 Rel []
Colors of variables: wff setvar class
Syntax hints:  wpss 3724  Rel wrel 5254   [] crpss 7083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-opab 4847  df-xp 5255  df-rel 5256  df-rpss 7084
This theorem is referenced by:  brrpssg  7086  compssiso  9398
  Copyright terms: Public domain W3C validator