Theorem psshepw 43752

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.

Step	Hyp	Ref	Expression
1		dfhe3 43739	. 2 ⊢ (◡ [⊊] hereditary 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → ∀𝑦(𝑥◡ [⊊] 𝑦 → 𝑦 ∈ 𝒫 𝐴)))
2		sstr2 4015	. . . . 5 ⊢ (𝑦 ⊆ 𝑥 → (𝑥 ⊆ 𝐴 → 𝑦 ⊆ 𝐴))
3		pssss 4121	. . . . 5 ⊢ (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝑥)
4	2, 3	syl11 33	. . . 4 ⊢ (𝑥 ⊆ 𝐴 → (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝐴))
5	4	alrimiv 1926	. . 3 ⊢ (𝑥 ⊆ 𝐴 → ∀𝑦(𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝐴))
6		velpw 4627	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
7		vex 3492	. . . . . . 7 ⊢ 𝑥 ∈ V
8		vex 3492	. . . . . . 7 ⊢ 𝑦 ∈ V
9	7, 8	brcnv 5907	. . . . . 6 ⊢ (𝑥◡ [⊊] 𝑦 ↔ 𝑦 [⊊] 𝑥)
10	7	brrpss 7763	. . . . . 6 ⊢ (𝑦 [⊊] 𝑥 ↔ 𝑦 ⊊ 𝑥)
11	9, 10	bitri 275	. . . . 5 ⊢ (𝑥◡ [⊊] 𝑦 ↔ 𝑦 ⊊ 𝑥)
12		velpw 4627	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
13	11, 12	imbi12i 350	. . . 4 ⊢ ((𝑥◡ [⊊] 𝑦 → 𝑦 ∈ 𝒫 𝐴) ↔ (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝐴))
14	13	albii 1817	. . 3 ⊢ (∀𝑦(𝑥◡ [⊊] 𝑦 → 𝑦 ∈ 𝒫 𝐴) ↔ ∀𝑦(𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝐴))
15	5, 6, 14	3imtr4i 292	. 2 ⊢ (𝑥 ∈ 𝒫 𝐴 → ∀𝑦(𝑥◡ [⊊] 𝑦 → 𝑦 ∈ 𝒫 𝐴))
16	1, 15	mpgbir 1797	1 ⊢ ◡ [⊊] hereditary 𝒫 𝐴

Syntax hints:   → wi 4  ∀wal 1535   ∈ wcel 2108   ⊆ wss 3976   ⊊ wpss 3977  𝒫 cpw 4622   class class class wbr 5166  ^◡ccnv 5699   [⊊] crpss 7759   hereditary whe 43736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-rpss 7760  df-he 43737

This theorem is referenced by:  sshepw  43753