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 41727
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 41726 . 2 [] hereditary 𝒫 𝐴
2 idhe 41725 . 2 I hereditary 𝒫 𝐴
3 unhe1 41723 . 2 (( [] hereditary 𝒫 𝐴 ∧ I hereditary 𝒫 𝐴) → ( [] ∪ I ) hereditary 𝒫 𝐴)
41, 2, 3mp2an 689 1 ( [] ∪ I ) hereditary 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  cun 3896  𝒫 cpw 4547   I cid 5517  ccnv 5619   [] crpss 7637   hereditary whe 41710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-br 5093  df-opab 5155  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-rpss 7638  df-he 41711
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator