Theorem psshepw 43777

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 43764	. 2 ⊢ (◡ [⊊] hereditary 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → ∀𝑦(𝑥◡ [⊊] 𝑦 → 𝑦 ∈ 𝒫 𝐴)))
2		sstr2 3953	. . . . 5 ⊢ (𝑦 ⊆ 𝑥 → (𝑥 ⊆ 𝐴 → 𝑦 ⊆ 𝐴))
3		pssss 4061	. . . . 5 ⊢ (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝑥)
4	2, 3	syl11 33	. . . 4 ⊢ (𝑥 ⊆ 𝐴 → (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝐴))
5	4	alrimiv 1927	. . 3 ⊢ (𝑥 ⊆ 𝐴 → ∀𝑦(𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝐴))
6		velpw 4568	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
7		vex 3451	. . . . . . 7 ⊢ 𝑥 ∈ V
8		vex 3451	. . . . . . 7 ⊢ 𝑦 ∈ V
9	7, 8	brcnv 5846	. . . . . 6 ⊢ (𝑥◡ [⊊] 𝑦 ↔ 𝑦 [⊊] 𝑥)
10	7	brrpss 7702	. . . . . 6 ⊢ (𝑦 [⊊] 𝑥 ↔ 𝑦 ⊊ 𝑥)
11	9, 10	bitri 275	. . . . 5 ⊢ (𝑥◡ [⊊] 𝑦 ↔ 𝑦 ⊊ 𝑥)
12		velpw 4568	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
13	11, 12	imbi12i 350	. . . 4 ⊢ ((𝑥◡ [⊊] 𝑦 → 𝑦 ∈ 𝒫 𝐴) ↔ (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝐴))
14	13	albii 1819	. . 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 1799	1 ⊢ ◡ [⊊] hereditary 𝒫 𝐴

Syntax hints:   → wi 4  ∀wal 1538   ∈ wcel 2109   ⊆ wss 3914   ⊊ wpss 3915  𝒫 cpw 4563   class class class wbr 5107  ^◡ccnv 5637   [⊊] crpss 7698   hereditary whe 43761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-rpss 7699  df-he 43762

This theorem is referenced by:  sshepw  43778