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Theorem psshepw 42040
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 42027 . 2 ( [] hereditary 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → ∀𝑦(𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴)))
2 sstr2 3950 . . . . 5 (𝑦𝑥 → (𝑥𝐴𝑦𝐴))
3 pssss 4054 . . . . 5 (𝑦𝑥𝑦𝑥)
42, 3syl11 33 . . . 4 (𝑥𝐴 → (𝑦𝑥𝑦𝐴))
54alrimiv 1930 . . 3 (𝑥𝐴 → ∀𝑦(𝑦𝑥𝑦𝐴))
6 velpw 4564 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
7 vex 3448 . . . . . . 7 𝑥 ∈ V
8 vex 3448 . . . . . . 7 𝑦 ∈ V
97, 8brcnv 5837 . . . . . 6 (𝑥 [] 𝑦𝑦 [] 𝑥)
107brrpss 7660 . . . . . 6 (𝑦 [] 𝑥𝑦𝑥)
119, 10bitri 274 . . . . 5 (𝑥 [] 𝑦𝑦𝑥)
12 velpw 4564 . . . . 5 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
1311, 12imbi12i 350 . . . 4 ((𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴) ↔ (𝑦𝑥𝑦𝐴))
1413albii 1821 . . 3 (∀𝑦(𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴) ↔ ∀𝑦(𝑦𝑥𝑦𝐴))
155, 6, 143imtr4i 291 . 2 (𝑥 ∈ 𝒫 𝐴 → ∀𝑦(𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴))
161, 15mpgbir 1801 1 [] hereditary 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1539  wcel 2106  wss 3909  wpss 3910  𝒫 cpw 4559   class class class wbr 5104  ccnv 5631   [] crpss 7656   hereditary whe 42024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-rpss 7657  df-he 42025
This theorem is referenced by:  sshepw  42041
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