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

Theorem sshepw 40013
Description: The relation between sets and their subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.)
Assertion
Ref Expression
sshepw ( [] ∪ I ) hereditary 𝒫 𝐴

Proof of Theorem sshepw
StepHypRef Expression
1 psshepw 40012 . 2 [] hereditary 𝒫 𝐴
2 idhe 40011 . 2 I hereditary 𝒫 𝐴
3 unhe1 40009 . 2 (( [] hereditary 𝒫 𝐴 ∧ I hereditary 𝒫 𝐴) → ( [] ∪ I ) hereditary 𝒫 𝐴)
41, 2, 3mp2an 688 1 ( [] ∪ I ) hereditary 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  cun 3931  𝒫 cpw 4535   I cid 5452  ccnv 5547   [] crpss 7437   hereditary whe 39996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-rpss 7438  df-he 39997
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator