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Theorem elpwdifsn 4713
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 1129 . . . . . 6 ((𝑆𝑊𝑆𝑉𝐴𝑆) → 𝑆𝑉)
21sselda 3964 . . . . 5 (((𝑆𝑊𝑆𝑉𝐴𝑆) ∧ 𝑥𝑆) → 𝑥𝑉)
3 df-nel 3121 . . . . . . . . 9 (𝐴𝑆 ↔ ¬ 𝐴𝑆)
43biimpi 217 . . . . . . . 8 (𝐴𝑆 → ¬ 𝐴𝑆)
543ad2ant3 1127 . . . . . . 7 ((𝑆𝑊𝑆𝑉𝐴𝑆) → ¬ 𝐴𝑆)
65anim1ci 615 . . . . . 6 (((𝑆𝑊𝑆𝑉𝐴𝑆) ∧ 𝑥𝑆) → (𝑥𝑆 ∧ ¬ 𝐴𝑆))
7 nelne2 3112 . . . . . 6 ((𝑥𝑆 ∧ ¬ 𝐴𝑆) → 𝑥𝐴)
86, 7syl 17 . . . . 5 (((𝑆𝑊𝑆𝑉𝐴𝑆) ∧ 𝑥𝑆) → 𝑥𝐴)
9 eldifsn 4711 . . . . 5 (𝑥 ∈ (𝑉 ∖ {𝐴}) ↔ (𝑥𝑉𝑥𝐴))
102, 8, 9sylanbrc 583 . . . 4 (((𝑆𝑊𝑆𝑉𝐴𝑆) ∧ 𝑥𝑆) → 𝑥 ∈ (𝑉 ∖ {𝐴}))
1110ex 413 . . 3 ((𝑆𝑊𝑆𝑉𝐴𝑆) → (𝑥𝑆𝑥 ∈ (𝑉 ∖ {𝐴})))
1211ssrdv 3970 . 2 ((𝑆𝑊𝑆𝑉𝐴𝑆) → 𝑆 ⊆ (𝑉 ∖ {𝐴}))
13 elpwg 4541 . . 3 (𝑆𝑊 → (𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}) ↔ 𝑆 ⊆ (𝑉 ∖ {𝐴})))
14133ad2ant1 1125 . 2 ((𝑆𝑊𝑆𝑉𝐴𝑆) → (𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}) ↔ 𝑆 ⊆ (𝑉 ∖ {𝐴})))
1512, 14mpbird 258 1 ((𝑆𝑊𝑆𝑉𝐴𝑆) → 𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079  wcel 2105  wne 3013  wnel 3120  cdif 3930  wss 3933  𝒫 cpw 4535  {csn 4557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-v 3494  df-dif 3936  df-in 3940  df-ss 3949  df-pw 4537  df-sn 4558
This theorem is referenced by:  uhgrspan1  27012  upgrreslem  27013  umgrreslem  27014  umgrres1lem  27019  upgrres1  27022
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