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Theorem psshepw 38603
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 38590 . 2 ( [] hereditary 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → ∀𝑦(𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴)))
2 sstr2 3752 . . . . 5 (𝑦𝑥 → (𝑥𝐴𝑦𝐴))
3 pssss 3845 . . . . 5 (𝑦𝑥𝑦𝑥)
42, 3syl11 33 . . . 4 (𝑥𝐴 → (𝑦𝑥𝑦𝐴))
54alrimiv 2005 . . 3 (𝑥𝐴 → ∀𝑦(𝑦𝑥𝑦𝐴))
6 selpw 4310 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
7 vex 3344 . . . . . . 7 𝑥 ∈ V
8 vex 3344 . . . . . . 7 𝑦 ∈ V
97, 8brcnv 5461 . . . . . 6 (𝑥 [] 𝑦𝑦 [] 𝑥)
107brrpss 7107 . . . . . 6 (𝑦 [] 𝑥𝑦𝑥)
119, 10bitri 264 . . . . 5 (𝑥 [] 𝑦𝑦𝑥)
12 selpw 4310 . . . . 5 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
1311, 12imbi12i 339 . . . 4 ((𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴) ↔ (𝑦𝑥𝑦𝐴))
1413albii 1896 . . 3 (∀𝑦(𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴) ↔ ∀𝑦(𝑦𝑥𝑦𝐴))
155, 6, 143imtr4i 281 . 2 (𝑥 ∈ 𝒫 𝐴 → ∀𝑦(𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴))
161, 15mpgbir 1875 1 [] hereditary 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1630  wcel 2140  wss 3716  wpss 3717  𝒫 cpw 4303   class class class wbr 4805  ccnv 5266   [] crpss 7103   hereditary whe 38587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-br 4806  df-opab 4866  df-xp 5273  df-rel 5274  df-cnv 5275  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-rpss 7104  df-he 38588
This theorem is referenced by:  sshepw  38604
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