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Theorem elpwdifsn 4791
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 1135 . . . . . 6 ((𝑆𝑊𝑆𝑉𝐴𝑆) → 𝑆𝑉)
21sselda 3981 . . . . 5 (((𝑆𝑊𝑆𝑉𝐴𝑆) ∧ 𝑥𝑆) → 𝑥𝑉)
3 df-nel 3045 . . . . . . . . 9 (𝐴𝑆 ↔ ¬ 𝐴𝑆)
43biimpi 215 . . . . . . . 8 (𝐴𝑆 → ¬ 𝐴𝑆)
543ad2ant3 1133 . . . . . . 7 ((𝑆𝑊𝑆𝑉𝐴𝑆) → ¬ 𝐴𝑆)
65anim1ci 614 . . . . . 6 (((𝑆𝑊𝑆𝑉𝐴𝑆) ∧ 𝑥𝑆) → (𝑥𝑆 ∧ ¬ 𝐴𝑆))
7 nelne2 3038 . . . . . 6 ((𝑥𝑆 ∧ ¬ 𝐴𝑆) → 𝑥𝐴)
86, 7syl 17 . . . . 5 (((𝑆𝑊𝑆𝑉𝐴𝑆) ∧ 𝑥𝑆) → 𝑥𝐴)
9 eldifsn 4789 . . . . 5 (𝑥 ∈ (𝑉 ∖ {𝐴}) ↔ (𝑥𝑉𝑥𝐴))
102, 8, 9sylanbrc 581 . . . 4 (((𝑆𝑊𝑆𝑉𝐴𝑆) ∧ 𝑥𝑆) → 𝑥 ∈ (𝑉 ∖ {𝐴}))
1110ex 411 . . 3 ((𝑆𝑊𝑆𝑉𝐴𝑆) → (𝑥𝑆𝑥 ∈ (𝑉 ∖ {𝐴})))
1211ssrdv 3987 . 2 ((𝑆𝑊𝑆𝑉𝐴𝑆) → 𝑆 ⊆ (𝑉 ∖ {𝐴}))
13 elpwg 4604 . . 3 (𝑆𝑊 → (𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}) ↔ 𝑆 ⊆ (𝑉 ∖ {𝐴})))
14133ad2ant1 1131 . 2 ((𝑆𝑊𝑆𝑉𝐴𝑆) → (𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}) ↔ 𝑆 ⊆ (𝑉 ∖ {𝐴})))
1512, 14mpbird 256 1 ((𝑆𝑊𝑆𝑉𝐴𝑆) → 𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1085  wcel 2104  wne 2938  wnel 3044  cdif 3944  wss 3947  𝒫 cpw 4601  {csn 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1087  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-nel 3045  df-v 3474  df-dif 3950  df-in 3954  df-ss 3964  df-pw 4603  df-sn 4628
This theorem is referenced by:  uhgrspan1  28827  upgrreslem  28828  umgrreslem  28829  umgrres1lem  28834  upgrres1  28837
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