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Theorem elpwdifsn 4794
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.)
Assertion
Ref Expression
elpwdifsn ((𝑆𝑊𝑆𝑉𝐴𝑆) → 𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}))

Proof of Theorem elpwdifsn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simp2 1136 . . . . . 6 ((𝑆𝑊𝑆𝑉𝐴𝑆) → 𝑆𝑉)
21sselda 3995 . . . . 5 (((𝑆𝑊𝑆𝑉𝐴𝑆) ∧ 𝑥𝑆) → 𝑥𝑉)
3 df-nel 3045 . . . . . . . . 9 (𝐴𝑆 ↔ ¬ 𝐴𝑆)
43biimpi 216 . . . . . . . 8 (𝐴𝑆 → ¬ 𝐴𝑆)
543ad2ant3 1134 . . . . . . 7 ((𝑆𝑊𝑆𝑉𝐴𝑆) → ¬ 𝐴𝑆)
65anim1ci 616 . . . . . 6 (((𝑆𝑊𝑆𝑉𝐴𝑆) ∧ 𝑥𝑆) → (𝑥𝑆 ∧ ¬ 𝐴𝑆))
7 nelne2 3038 . . . . . 6 ((𝑥𝑆 ∧ ¬ 𝐴𝑆) → 𝑥𝐴)
86, 7syl 17 . . . . 5 (((𝑆𝑊𝑆𝑉𝐴𝑆) ∧ 𝑥𝑆) → 𝑥𝐴)
9 eldifsn 4791 . . . . 5 (𝑥 ∈ (𝑉 ∖ {𝐴}) ↔ (𝑥𝑉𝑥𝐴))
102, 8, 9sylanbrc 583 . . . 4 (((𝑆𝑊𝑆𝑉𝐴𝑆) ∧ 𝑥𝑆) → 𝑥 ∈ (𝑉 ∖ {𝐴}))
1110ex 412 . . 3 ((𝑆𝑊𝑆𝑉𝐴𝑆) → (𝑥𝑆𝑥 ∈ (𝑉 ∖ {𝐴})))
1211ssrdv 4001 . 2 ((𝑆𝑊𝑆𝑉𝐴𝑆) → 𝑆 ⊆ (𝑉 ∖ {𝐴}))
13 elpwg 4608 . . 3 (𝑆𝑊 → (𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}) ↔ 𝑆 ⊆ (𝑉 ∖ {𝐴})))
14133ad2ant1 1132 . 2 ((𝑆𝑊𝑆𝑉𝐴𝑆) → (𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}) ↔ 𝑆 ⊆ (𝑉 ∖ {𝐴})))
1512, 14mpbird 257 1 ((𝑆𝑊𝑆𝑉𝐴𝑆) → 𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086  wcel 2106  wne 2938  wnel 3044  cdif 3960  wss 3963  𝒫 cpw 4605  {csn 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-nel 3045  df-v 3480  df-dif 3966  df-ss 3980  df-pw 4607  df-sn 4632
This theorem is referenced by:  uhgrspan1  29335  upgrreslem  29336  umgrreslem  29337  umgrres1lem  29342  upgrres1  29345
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