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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 1132 | . . . . . 6 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → 𝑆 ⊆ 𝑉) | |
2 | 1 | sselda 3745 | . . . . 5 ⊢ (((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑉) |
3 | df-nel 3037 | . . . . . . . . . 10 ⊢ (𝐴 ∉ 𝑆 ↔ ¬ 𝐴 ∈ 𝑆) | |
4 | 3 | biimpi 206 | . . . . . . . . 9 ⊢ (𝐴 ∉ 𝑆 → ¬ 𝐴 ∈ 𝑆) |
5 | 4 | 3ad2ant3 1130 | . . . . . . . 8 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → ¬ 𝐴 ∈ 𝑆) |
6 | 5 | anim1i 593 | . . . . . . 7 ⊢ (((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) ∧ 𝑥 ∈ 𝑆) → (¬ 𝐴 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆)) |
7 | 6 | ancomd 466 | . . . . . 6 ⊢ (((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑥 ∈ 𝑆 ∧ ¬ 𝐴 ∈ 𝑆)) |
8 | nelne2 3030 | . . . . . 6 ⊢ ((𝑥 ∈ 𝑆 ∧ ¬ 𝐴 ∈ 𝑆) → 𝑥 ≠ 𝐴) | |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ (((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) ∧ 𝑥 ∈ 𝑆) → 𝑥 ≠ 𝐴) |
10 | eldifsn 4463 | . . . . 5 ⊢ (𝑥 ∈ (𝑉 ∖ {𝐴}) ↔ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 𝐴)) | |
11 | 2, 9, 10 | sylanbrc 701 | . . . 4 ⊢ (((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (𝑉 ∖ {𝐴})) |
12 | 11 | ex 449 | . . 3 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → (𝑥 ∈ 𝑆 → 𝑥 ∈ (𝑉 ∖ {𝐴}))) |
13 | 12 | ssrdv 3751 | . 2 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → 𝑆 ⊆ (𝑉 ∖ {𝐴})) |
14 | elpwg 4311 | . . 3 ⊢ (𝑆 ∈ 𝑊 → (𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}) ↔ 𝑆 ⊆ (𝑉 ∖ {𝐴}))) | |
15 | 14 | 3ad2ant1 1128 | . 2 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → (𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}) ↔ 𝑆 ⊆ (𝑉 ∖ {𝐴}))) |
16 | 13, 15 | mpbird 247 | 1 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → 𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1072 ∈ wcel 2140 ≠ wne 2933 ∉ wnel 3036 ∖ cdif 3713 ⊆ wss 3716 𝒫 cpw 4303 {csn 4322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1989 ax-6 2055 ax-7 2091 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2048 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ne 2934 df-nel 3037 df-v 3343 df-dif 3719 df-in 3723 df-ss 3730 df-pw 4305 df-sn 4323 |
This theorem is referenced by: uhgrspan1 26416 upgrreslem 26417 umgrreslem 26418 umgrres1lem 26423 upgrres1 26426 |
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