Theorem sshepw 45050

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 45049	. 2 ⊢ ◡ [⊊] hereditary 𝒫 𝐴
2		idhe 45048	. 2 ⊢ I hereditary 𝒫 𝐴
3		unhe1 45046	. 2 ⊢ ((◡ [⊊] hereditary 𝒫 𝐴 ∧ I hereditary 𝒫 𝐴) → (◡ [⊊] ∪ I ) hereditary 𝒫 𝐴)
4	1, 2, 3	mp2an 705	1 ⊢ (◡ [⊊] ∪ I ) hereditary 𝒫 𝐴

This proof depends on syntax axioms:   ∪ cun 4152  𝒫 cpw 4560   I cid 5552  ^◡ccnv 5657   [⊊] crpss 7770   hereditary whe 45033

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-rpss 7771  df-he 45034

This theorem is used by: (None)