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Mirrors > Home > MPE Home > Th. List > Mathboxes > psshepw | Structured version Visualization version GIF version |
Description: The relation between sets and their proper subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.) |
Ref | Expression |
---|---|
psshepw | ⊢ ◡ [⊊] hereditary 𝒫 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfhe3 43479 | . 2 ⊢ (◡ [⊊] hereditary 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → ∀𝑦(𝑥◡ [⊊] 𝑦 → 𝑦 ∈ 𝒫 𝐴))) | |
2 | sstr2 3985 | . . . . 5 ⊢ (𝑦 ⊆ 𝑥 → (𝑥 ⊆ 𝐴 → 𝑦 ⊆ 𝐴)) | |
3 | pssss 4091 | . . . . 5 ⊢ (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝑥) | |
4 | 2, 3 | syl11 33 | . . . 4 ⊢ (𝑥 ⊆ 𝐴 → (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝐴)) |
5 | 4 | alrimiv 1923 | . . 3 ⊢ (𝑥 ⊆ 𝐴 → ∀𝑦(𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝐴)) |
6 | velpw 4602 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
7 | vex 3466 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
8 | vex 3466 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | brcnv 5881 | . . . . . 6 ⊢ (𝑥◡ [⊊] 𝑦 ↔ 𝑦 [⊊] 𝑥) |
10 | 7 | brrpss 7729 | . . . . . 6 ⊢ (𝑦 [⊊] 𝑥 ↔ 𝑦 ⊊ 𝑥) |
11 | 9, 10 | bitri 274 | . . . . 5 ⊢ (𝑥◡ [⊊] 𝑦 ↔ 𝑦 ⊊ 𝑥) |
12 | velpw 4602 | . . . . 5 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴) | |
13 | 11, 12 | imbi12i 349 | . . . 4 ⊢ ((𝑥◡ [⊊] 𝑦 → 𝑦 ∈ 𝒫 𝐴) ↔ (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝐴)) |
14 | 13 | albii 1814 | . . 3 ⊢ (∀𝑦(𝑥◡ [⊊] 𝑦 → 𝑦 ∈ 𝒫 𝐴) ↔ ∀𝑦(𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝐴)) |
15 | 5, 6, 14 | 3imtr4i 291 | . 2 ⊢ (𝑥 ∈ 𝒫 𝐴 → ∀𝑦(𝑥◡ [⊊] 𝑦 → 𝑦 ∈ 𝒫 𝐴)) |
16 | 1, 15 | mpgbir 1794 | 1 ⊢ ◡ [⊊] hereditary 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1532 ∈ wcel 2099 ⊆ wss 3946 ⊊ wpss 3947 𝒫 cpw 4597 class class class wbr 5145 ◡ccnv 5673 [⊊] crpss 7725 hereditary whe 43476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-11 2147 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-br 5146 df-opab 5208 df-xp 5680 df-rel 5681 df-cnv 5682 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-rpss 7726 df-he 43477 |
This theorem is referenced by: sshepw 43493 |
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