Theorem elpwdifsn 40220

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 1054	. . . . . 6 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → 𝑆 ⊆ 𝑉)
2	1	sselda 3472	. . . . 5 ⊢ (((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑉)
3		df-nel 2687	. . . . . . . . . 10 ⊢ (𝐴 ∉ 𝑆 ↔ ¬ 𝐴 ∈ 𝑆)
4	3	biimpi 204	. . . . . . . . 9 ⊢ (𝐴 ∉ 𝑆 → ¬ 𝐴 ∈ 𝑆)
5	4	3ad2ant3 1076	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → ¬ 𝐴 ∈ 𝑆)
6	5	anim1i 589	. . . . . . 7 ⊢ (((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) ∧ 𝑥 ∈ 𝑆) → (¬ 𝐴 ∈ 𝑆 ∧ 𝑥 ∈ 𝑆))
7	6	ancomd 465	. . . . . 6 ⊢ (((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑥 ∈ 𝑆 ∧ ¬ 𝐴 ∈ 𝑆))
8		nelne2 2783	. . . . . 6 ⊢ ((𝑥 ∈ 𝑆 ∧ ¬ 𝐴 ∈ 𝑆) → 𝑥 ≠ 𝐴)
9	7, 8	syl 17	. . . . 5 ⊢ (((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) ∧ 𝑥 ∈ 𝑆) → 𝑥 ≠ 𝐴)
10		eldifsn 4163	. . . . 5 ⊢ (𝑥 ∈ (𝑉 ∖ {𝐴}) ↔ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 𝐴))
11	2, 9, 10	sylanbrc 694	. . . 4 ⊢ (((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (𝑉 ∖ {𝐴}))
12	11	ex 448	. . 3 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → (𝑥 ∈ 𝑆 → 𝑥 ∈ (𝑉 ∖ {𝐴})))
13	12	ssrdv 3478	. 2 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → 𝑆 ⊆ (𝑉 ∖ {𝐴}))
14		elpwg 4019	. . 3 ⊢ (𝑆 ∈ 𝑊 → (𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}) ↔ 𝑆 ⊆ (𝑉 ∖ {𝐴})))
15	14	3ad2ant1 1074	. 2 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → (𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}) ↔ 𝑆 ⊆ (𝑉 ∖ {𝐴})))
16	13, 15	mpbird 245	1 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ∧ 𝐴 ∉ 𝑆) → 𝑆 ∈ 𝒫 (𝑉 ∖ {𝐴}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 194   ∧ wa 382   ∧ w3a 1030   ∈ wcel 1938   ≠ wne 2684   ∉ wnel 2685   ∖ cdif 3441   ⊆ wss 3444  𝒫 cpw 4011  {csn 4028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494

This theorem depends on definitions:  df-bi 195  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-v 3079  df-dif 3447  df-in 3451  df-ss 3458  df-pw 4013  df-sn 4029

This theorem is referenced by:  uhgrspan1  40632  umgrres1lem  40634  upgrres1  40637