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Theorem elpwdifsn 4784
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 1134 . . . . . 6 ((𝑆𝑊𝑆𝑉𝐴𝑆) → 𝑆𝑉)
21sselda 3974 . . . . 5 (((𝑆𝑊𝑆𝑉𝐴𝑆) ∧ 𝑥𝑆) → 𝑥𝑉)
3 df-nel 3039 . . . . . . . . 9 (𝐴𝑆 ↔ ¬ 𝐴𝑆)
43biimpi 215 . . . . . . . 8 (𝐴𝑆 → ¬ 𝐴𝑆)
543ad2ant3 1132 . . . . . . 7 ((𝑆𝑊𝑆𝑉𝐴𝑆) → ¬ 𝐴𝑆)
65anim1ci 615 . . . . . 6 (((𝑆𝑊𝑆𝑉𝐴𝑆) ∧ 𝑥𝑆) → (𝑥𝑆 ∧ ¬ 𝐴𝑆))
7 nelne2 3032 . . . . . 6 ((𝑥𝑆 ∧ ¬ 𝐴𝑆) → 𝑥𝐴)
86, 7syl 17 . . . . 5 (((𝑆𝑊𝑆𝑉𝐴𝑆) ∧ 𝑥𝑆) → 𝑥𝐴)
9 eldifsn 4782 . . . . 5 (𝑥 ∈ (𝑉 ∖ {𝐴}) ↔ (𝑥𝑉𝑥𝐴))
102, 8, 9sylanbrc 582 . . . 4 (((𝑆𝑊𝑆𝑉𝐴𝑆) ∧ 𝑥𝑆) → 𝑥 ∈ (𝑉 ∖ {𝐴}))
1110ex 412 . . 3 ((𝑆𝑊𝑆𝑉𝐴𝑆) → (𝑥𝑆𝑥 ∈ (𝑉 ∖ {𝐴})))
1211ssrdv 3980 . 2 ((𝑆𝑊𝑆𝑉𝐴𝑆) → 𝑆 ⊆ (𝑉 ∖ {𝐴}))
13 elpwg 4597 . . 3 (𝑆𝑊 → (𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}) ↔ 𝑆 ⊆ (𝑉 ∖ {𝐴})))
14133ad2ant1 1130 . 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 205  wa 395  w3a 1084  wcel 2098  wne 2932  wnel 3038  cdif 3937  wss 3940  𝒫 cpw 4594  {csn 4620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-nel 3039  df-v 3468  df-dif 3943  df-in 3947  df-ss 3957  df-pw 4596  df-sn 4621
This theorem is referenced by:  uhgrspan1  28995  upgrreslem  28996  umgrreslem  28997  umgrres1lem  29002  upgrres1  29005
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