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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 37089 | . 2 ⊢ (◡ [⊊] hereditary 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → ∀𝑦(𝑥◡ [⊊] 𝑦 → 𝑦 ∈ 𝒫 𝐴))) | |
2 | pssss 3664 | . . . . . 6 ⊢ (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝑥) | |
3 | sstr2 3575 | . . . . . 6 ⊢ (𝑦 ⊆ 𝑥 → (𝑥 ⊆ 𝐴 → 𝑦 ⊆ 𝐴)) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝑦 ⊊ 𝑥 → (𝑥 ⊆ 𝐴 → 𝑦 ⊆ 𝐴)) |
5 | 4 | com12 32 | . . . 4 ⊢ (𝑥 ⊆ 𝐴 → (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝐴)) |
6 | 5 | alrimiv 1842 | . . 3 ⊢ (𝑥 ⊆ 𝐴 → ∀𝑦(𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝐴)) |
7 | selpw 4115 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
8 | vex 3176 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
9 | vex 3176 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | brcnv 5227 | . . . . . 6 ⊢ (𝑥◡ [⊊] 𝑦 ↔ 𝑦 [⊊] 𝑥) |
11 | 8 | brrpss 6838 | . . . . . 6 ⊢ (𝑦 [⊊] 𝑥 ↔ 𝑦 ⊊ 𝑥) |
12 | 10, 11 | bitri 263 | . . . . 5 ⊢ (𝑥◡ [⊊] 𝑦 ↔ 𝑦 ⊊ 𝑥) |
13 | selpw 4115 | . . . . 5 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴) | |
14 | 12, 13 | imbi12i 339 | . . . 4 ⊢ ((𝑥◡ [⊊] 𝑦 → 𝑦 ∈ 𝒫 𝐴) ↔ (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝐴)) |
15 | 14 | albii 1737 | . . 3 ⊢ (∀𝑦(𝑥◡ [⊊] 𝑦 → 𝑦 ∈ 𝒫 𝐴) ↔ ∀𝑦(𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝐴)) |
16 | 6, 7, 15 | 3imtr4i 280 | . 2 ⊢ (𝑥 ∈ 𝒫 𝐴 → ∀𝑦(𝑥◡ [⊊] 𝑦 → 𝑦 ∈ 𝒫 𝐴)) |
17 | 1, 16 | mpgbir 1717 | 1 ⊢ ◡ [⊊] hereditary 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 ∈ wcel 1977 ⊆ wss 3540 ⊊ wpss 3541 𝒫 cpw 4108 class class class wbr 4583 ◡ccnv 5037 [⊊] crpss 6834 hereditary whe 37086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-rpss 6835 df-he 37087 |
This theorem is referenced by: sshepw 37103 |
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