Theorem elpwdifsn 4785

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	⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → 𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}))

Proof of Theorem elpwdifsn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1137	. . . . . 6 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → 𝑆 ⊆ 𝑉)
2	1	sselda 3978	. . . . 5 ⊢ (((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑉)
3		df-nel 3046	. . . . . . . . 9 ⊢ (𝐴 ∉ 𝑆 ↔ ¬ 𝐴 ∈ 𝑆)
4	3	biimpi 215	. . . . . . . 8 ⊢ (𝐴 ∉ 𝑆 → ¬ 𝐴 ∈ 𝑆)
5	4	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → ¬ 𝐴 ∈ 𝑆)
6	5	anim1ci 616	. . . . . 6 ⊢ (((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑥 ∈ 𝑆 ∧ ¬ 𝐴 ∈ 𝑆))
7		nelne2 3039	. . . . . 6 ⊢ ((𝑥 ∈ 𝑆 ∧ ¬ 𝐴 ∈ 𝑆) → 𝑥 ≠ 𝐴)
8	6, 7	syl 17	. . . . 5 ⊢ (((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) ∧ 𝑥 ∈ 𝑆) → 𝑥 ≠ 𝐴)
9		eldifsn 4783	. . . . 5 ⊢ (𝑥 ∈ (𝑉 ∖ {𝐴}) ↔ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 𝐴))
10	2, 8, 9	sylanbrc 583	. . . 4 ⊢ (((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (𝑉 ∖ {𝐴}))
11	10	ex 413	. . 3 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → (𝑥 ∈ 𝑆 → 𝑥 ∈ (𝑉 ∖ {𝐴})))
12	11	ssrdv 3984	. 2 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → 𝑆 ⊆ (𝑉 ∖ {𝐴}))
13		elpwg 4599	. . 3 ⊢ (𝑆 ∈ 𝑊 → (𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}) ↔ 𝑆 ⊆ (𝑉 ∖ {𝐴})))
14	13	3ad2ant1 1133	. 2 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → (𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}) ↔ 𝑆 ⊆ (𝑉 ∖ {𝐴})))
15	12, 14	mpbird 256	1 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → 𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   ∈ wcel 2106   ≠ wne 2939   ∉ wnel 3045   ∖ cdif 3941   ⊆ wss 3944  𝒫 cpw 4596  {csn 4622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-nel 3046  df-v 3475  df-dif 3947  df-in 3951  df-ss 3961  df-pw 4598  df-sn 4623

This theorem is referenced by:  uhgrspan1  28425  upgrreslem  28426  umgrreslem  28427  umgrres1lem  28432  upgrres1  28435