Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  0he Structured version   Visualization version   GIF version

Theorem 0he 43772
Description: The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.)
Assertion
Ref Expression
0he ∅ hereditary 𝐴

Proof of Theorem 0he
StepHypRef Expression
1 0ima 6098 . . 3 (∅ “ 𝐴) = ∅
2 0ss 4406 . . 3 ∅ ⊆ 𝐴
31, 2eqsstri 4030 . 2 (∅ “ 𝐴) ⊆ 𝐴
4 df-he 43763 . 2 (∅ hereditary 𝐴 ↔ (∅ “ 𝐴) ⊆ 𝐴)
53, 4mpbir 231 1 ∅ hereditary 𝐴
Colors of variables: wff setvar class
Syntax hints:  wss 3963  c0 4339  cima 5692   hereditary whe 43762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-he 43763
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator