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Mirrors > Home > MPE Home > Th. List > elpwdifsn | Structured version Visualization version GIF version |
Description: A subset of a set is an element of the power set of the difference of the set with a singleton if the subset does not contain the singleton element. (Contributed by AV, 10-Jan-2020.) |
Ref | Expression |
---|---|
elpwdifsn | ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → 𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1138 | . . . . . 6 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → 𝑆 ⊆ 𝑉) | |
2 | 1 | sselda 3980 | . . . . 5 ⊢ (((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑉) |
3 | df-nel 3048 | . . . . . . . . 9 ⊢ (𝐴 ∉ 𝑆 ↔ ¬ 𝐴 ∈ 𝑆) | |
4 | 3 | biimpi 215 | . . . . . . . 8 ⊢ (𝐴 ∉ 𝑆 → ¬ 𝐴 ∈ 𝑆) |
5 | 4 | 3ad2ant3 1136 | . . . . . . 7 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → ¬ 𝐴 ∈ 𝑆) |
6 | 5 | anim1ci 617 | . . . . . 6 ⊢ (((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑥 ∈ 𝑆 ∧ ¬ 𝐴 ∈ 𝑆)) |
7 | nelne2 3041 | . . . . . 6 ⊢ ((𝑥 ∈ 𝑆 ∧ ¬ 𝐴 ∈ 𝑆) → 𝑥 ≠ 𝐴) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) ∧ 𝑥 ∈ 𝑆) → 𝑥 ≠ 𝐴) |
9 | eldifsn 4786 | . . . . 5 ⊢ (𝑥 ∈ (𝑉 ∖ {𝐴}) ↔ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 𝐴)) | |
10 | 2, 8, 9 | sylanbrc 584 | . . . 4 ⊢ (((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (𝑉 ∖ {𝐴})) |
11 | 10 | ex 414 | . . 3 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → (𝑥 ∈ 𝑆 → 𝑥 ∈ (𝑉 ∖ {𝐴}))) |
12 | 11 | ssrdv 3986 | . 2 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → 𝑆 ⊆ (𝑉 ∖ {𝐴})) |
13 | elpwg 4601 | . . 3 ⊢ (𝑆 ∈ 𝑊 → (𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}) ↔ 𝑆 ⊆ (𝑉 ∖ {𝐴}))) | |
14 | 13 | 3ad2ant1 1134 | . 2 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → (𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}) ↔ 𝑆 ⊆ (𝑉 ∖ {𝐴}))) |
15 | 12, 14 | mpbird 257 | 1 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → 𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 ≠ wne 2941 ∉ wnel 3047 ∖ cdif 3943 ⊆ wss 3946 𝒫 cpw 4598 {csn 4624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-3an 1090 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-nel 3048 df-v 3477 df-dif 3949 df-in 3953 df-ss 3963 df-pw 4600 df-sn 4625 |
This theorem is referenced by: uhgrspan1 28527 upgrreslem 28528 umgrreslem 28529 umgrres1lem 28534 upgrres1 28537 |
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