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Theorem sshepw 38918
 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 38917 . 2 [] hereditary 𝒫 𝐴
2 idhe 38916 . 2 I hereditary 𝒫 𝐴
3 unhe1 38914 . 2 (( [] hereditary 𝒫 𝐴 ∧ I hereditary 𝒫 𝐴) → ( [] ∪ I ) hereditary 𝒫 𝐴)
41, 2, 3mp2an 683 1 ( [] ∪ I ) hereditary 𝒫 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∪ cun 3796  𝒫 cpw 4380   I cid 5251  ◡ccnv 5345   [⊊] crpss 7201   hereditary whe 38901 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-rpss 7202  df-he 38902 This theorem is referenced by: (None)
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