Theorem fin23lem7 10356

Description: Lemma for isfin2-2 10359. The componentwise complement of a nonempty collection of sets is nonempty. (Contributed by Stefan O'Rear, 31-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)

Assertion

Ref	Expression
fin23lem7	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴 ∧ 𝐵 ≠ ∅) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑥) ∈ 𝐵} ≠ ∅)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem fin23lem7

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 4353	. . . 4 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
2		difss 4136	. . . . . . . 8 ⊢ (𝐴 ∖ 𝑦) ⊆ 𝐴
3		elpw2g 5333	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ 𝑦) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ 𝑦) ⊆ 𝐴))
4	3	ad2antrr 726	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦 ∈ 𝐵) → ((𝐴 ∖ 𝑦) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ 𝑦) ⊆ 𝐴))
5	2, 4	mpbiri 258	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝐴 ∖ 𝑦) ∈ 𝒫 𝐴)
6		simpr 484	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) → 𝐵 ⊆ 𝒫 𝐴)
7	6	sselda 3983	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝒫 𝐴)
8	7	elpwid 4609	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ⊆ 𝐴)
9		dfss4 4269	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦)
10	8, 9	sylib 218	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦)
11		simpr 484	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
12	10, 11	eqeltrd 2841	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝐴 ∖ (𝐴 ∖ 𝑦)) ∈ 𝐵)
13		difeq2 4120	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 ∖ 𝑦) → (𝐴 ∖ 𝑥) = (𝐴 ∖ (𝐴 ∖ 𝑦)))
14	13	eleq1d 2826	. . . . . . . 8 ⊢ (𝑥 = (𝐴 ∖ 𝑦) → ((𝐴 ∖ 𝑥) ∈ 𝐵 ↔ (𝐴 ∖ (𝐴 ∖ 𝑦)) ∈ 𝐵))
15	14	rspcev 3622	. . . . . . 7 ⊢ (((𝐴 ∖ 𝑦) ∈ 𝒫 𝐴 ∧ (𝐴 ∖ (𝐴 ∖ 𝑦)) ∈ 𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐴 ∖ 𝑥) ∈ 𝐵)
16	5, 12, 15	syl2anc 584	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐴 ∖ 𝑥) ∈ 𝐵)
17	16	ex 412	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝒫 𝐴(𝐴 ∖ 𝑥) ∈ 𝐵))
18	17	exlimdv 1933	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) → (∃𝑦 𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝒫 𝐴(𝐴 ∖ 𝑥) ∈ 𝐵))
19	1, 18	biimtrid 242	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) → (𝐵 ≠ ∅ → ∃𝑥 ∈ 𝒫 𝐴(𝐴 ∖ 𝑥) ∈ 𝐵))
20	19	3impia 1118	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝒫 𝐴(𝐴 ∖ 𝑥) ∈ 𝐵)
21		rabn0 4389	. 2 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑥) ∈ 𝐵} ≠ ∅ ↔ ∃𝑥 ∈ 𝒫 𝐴(𝐴 ∖ 𝑥) ∈ 𝐵)
22	20, 21	sylibr 234	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴 ∧ 𝐵 ≠ ∅) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑥) ∈ 𝐵} ≠ ∅)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2940  ∃wrex 3070  {crab 3436   ∖ cdif 3948   ⊆ wss 3951  ∅c0 4333  𝒫 cpw 4600

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-in 3958  df-ss 3968  df-nul 4334  df-pw 4602

This theorem is referenced by:  fin2i2  10358  isfin2-2  10359