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Theorem psshepw 40018
Description: The relation between sets and their proper subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.)
Assertion
Ref Expression
psshepw [] hereditary 𝒫 𝐴

Proof of Theorem psshepw
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfhe3 40005 . 2 ( [] hereditary 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → ∀𝑦(𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴)))
2 sstr2 3978 . . . . 5 (𝑦𝑥 → (𝑥𝐴𝑦𝐴))
3 pssss 4076 . . . . 5 (𝑦𝑥𝑦𝑥)
42, 3syl11 33 . . . 4 (𝑥𝐴 → (𝑦𝑥𝑦𝐴))
54alrimiv 1921 . . 3 (𝑥𝐴 → ∀𝑦(𝑦𝑥𝑦𝐴))
6 velpw 4550 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
7 vex 3503 . . . . . . 7 𝑥 ∈ V
8 vex 3503 . . . . . . 7 𝑦 ∈ V
97, 8brcnv 5752 . . . . . 6 (𝑥 [] 𝑦𝑦 [] 𝑥)
107brrpss 7446 . . . . . 6 (𝑦 [] 𝑥𝑦𝑥)
119, 10bitri 276 . . . . 5 (𝑥 [] 𝑦𝑦𝑥)
12 velpw 4550 . . . . 5 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
1311, 12imbi12i 352 . . . 4 ((𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴) ↔ (𝑦𝑥𝑦𝐴))
1413albii 1813 . . 3 (∀𝑦(𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴) ↔ ∀𝑦(𝑦𝑥𝑦𝐴))
155, 6, 143imtr4i 293 . 2 (𝑥 ∈ 𝒫 𝐴 → ∀𝑦(𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴))
161, 15mpgbir 1793 1 [] hereditary 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1528  wcel 2107  wss 3940  wpss 3941  𝒫 cpw 4542   class class class wbr 5063  ccnv 5553   [] crpss 7442   hereditary whe 40002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-xp 5560  df-rel 5561  df-cnv 5562  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-rpss 7443  df-he 40003
This theorem is referenced by:  sshepw  40019
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