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Theorem elpwdifsn 4428
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 3709 . . . . 5 (((𝑆𝑊𝑆𝑉𝐴𝑆) ∧ 𝑥𝑆) → 𝑥𝑉)
3 df-nel 3000 . . . . . . . . . 10 (𝐴𝑆 ↔ ¬ 𝐴𝑆)
43biimpi 206 . . . . . . . . 9 (𝐴𝑆 → ¬ 𝐴𝑆)
543ad2ant3 1127 . . . . . . . 8 ((𝑆𝑊𝑆𝑉𝐴𝑆) → ¬ 𝐴𝑆)
65anim1i 593 . . . . . . 7 (((𝑆𝑊𝑆𝑉𝐴𝑆) ∧ 𝑥𝑆) → (¬ 𝐴𝑆𝑥𝑆))
76ancomd 466 . . . . . 6 (((𝑆𝑊𝑆𝑉𝐴𝑆) ∧ 𝑥𝑆) → (𝑥𝑆 ∧ ¬ 𝐴𝑆))
8 nelne2 2993 . . . . . 6 ((𝑥𝑆 ∧ ¬ 𝐴𝑆) → 𝑥𝐴)
97, 8syl 17 . . . . 5 (((𝑆𝑊𝑆𝑉𝐴𝑆) ∧ 𝑥𝑆) → 𝑥𝐴)
10 eldifsn 4425 . . . . 5 (𝑥 ∈ (𝑉 ∖ {𝐴}) ↔ (𝑥𝑉𝑥𝐴))
112, 9, 10sylanbrc 701 . . . 4 (((𝑆𝑊𝑆𝑉𝐴𝑆) ∧ 𝑥𝑆) → 𝑥 ∈ (𝑉 ∖ {𝐴}))
1211ex 449 . . 3 ((𝑆𝑊𝑆𝑉𝐴𝑆) → (𝑥𝑆𝑥 ∈ (𝑉 ∖ {𝐴})))
1312ssrdv 3715 . 2 ((𝑆𝑊𝑆𝑉𝐴𝑆) → 𝑆 ⊆ (𝑉 ∖ {𝐴}))
14 elpwg 4274 . . 3 (𝑆𝑊 → (𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}) ↔ 𝑆 ⊆ (𝑉 ∖ {𝐴})))
15143ad2ant1 1125 . 2 ((𝑆𝑊𝑆𝑉𝐴𝑆) → (𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}) ↔ 𝑆 ⊆ (𝑉 ∖ {𝐴})))
1613, 15mpbird 247 1 ((𝑆𝑊𝑆𝑉𝐴𝑆) → 𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072  wcel 2103  wne 2896  wnel 2999  cdif 3677  wss 3680  𝒫 cpw 4266  {csn 4285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-v 3306  df-dif 3683  df-in 3687  df-ss 3694  df-pw 4268  df-sn 4286
This theorem is referenced by:  uhgrspan1  26315  upgrreslem  26316  umgrreslem  26317  umgrres1lem  26322  upgrres1  26325
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