Theorem fin23lem7 10354

Description: Lemma for isfin2-2 10357. 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 4359	. . . 4 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
2		difss 4146	. . . . . . . 8 ⊢ (𝐴 ∖ 𝑦) ⊆ 𝐴
3		elpw2g 5339	. . . . . . . . 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 3995	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝒫 𝐴)
8	7	elpwid 4614	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ⊆ 𝐴)
9		dfss4 4275	. . . . . . . . 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 2839	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝐴 ∖ (𝐴 ∖ 𝑦)) ∈ 𝐵)
13		difeq2 4130	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 ∖ 𝑦) → (𝐴 ∖ 𝑥) = (𝐴 ∖ (𝐴 ∖ 𝑦)))
14	13	eleq1d 2824	. . . . . . . 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 1931	. . . 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 1116	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝒫 𝐴(𝐴 ∖ 𝑥) ∈ 𝐵)
21		rabn0 4395	. 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 1086   = wceq 1537  ∃wex 1776   ∈ wcel 2106   ≠ wne 2938  ∃wrex 3068  {crab 3433   ∖ cdif 3960   ⊆ wss 3963  ∅c0 4339  𝒫 cpw 4605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340  df-pw 4607

This theorem is referenced by:  fin2i2  10356  isfin2-2  10357