Theorem sshepw 43828

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

Step	Hyp	Ref	Expression
1		psshepw 43827	. 2 ⊢ ◡ [⊊] hereditary 𝒫 𝐴
2		idhe 43826	. 2 ⊢ I hereditary 𝒫 𝐴
3		unhe1 43824	. 2 ⊢ ((◡ [⊊] hereditary 𝒫 𝐴 ∧ I hereditary 𝒫 𝐴) → (◡ [⊊] ∪ I ) hereditary 𝒫 𝐴)
4	1, 2, 3	mp2an 692	1 ⊢ (◡ [⊊] ∪ I ) hereditary 𝒫 𝐴

Syntax hints:   ∪ cun 3900  𝒫 cpw 4550   I cid 5510  ^◡ccnv 5615   [⊊] crpss 7655   hereditary whe 43811

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-rpss 7656  df-he 43812

This theorem is referenced by: (None)