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 42523
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 6077 . . 3 (∅ “ 𝐴) = ∅
2 0ss 4396 . . 3 ∅ ⊆ 𝐴
31, 2eqsstri 4016 . 2 (∅ “ 𝐴) ⊆ 𝐴
4 df-he 42514 . 2 (∅ hereditary 𝐴 ↔ (∅ “ 𝐴) ⊆ 𝐴)
53, 4mpbir 230 1 ∅ hereditary 𝐴
Colors of variables: wff setvar class
Syntax hints:  wss 3948  c0 4322  cima 5679   hereditary whe 42513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-he 42514
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator