Theorem fin23lem7 9730

Description: Lemma for isfin2-2 9733. 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 4308	. . . 4 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
2		difss 4106	. . . . . . . 8 ⊢ (𝐴 ∖ 𝑦) ⊆ 𝐴
3		elpw2g 5238	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ 𝑦) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ 𝑦) ⊆ 𝐴))
4	3	ad2antrr 724	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦 ∈ 𝐵) → ((𝐴 ∖ 𝑦) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ 𝑦) ⊆ 𝐴))
5	2, 4	mpbiri 260	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝐴 ∖ 𝑦) ∈ 𝒫 𝐴)
6		simpr 487	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) → 𝐵 ⊆ 𝒫 𝐴)
7	6	sselda 3965	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝒫 𝐴)
8	7	elpwid 4551	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ⊆ 𝐴)
9		dfss4 4233	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦)
10	8, 9	sylib 220	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦)
11		simpr 487	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
12	10, 11	eqeltrd 2911	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝐴 ∖ (𝐴 ∖ 𝑦)) ∈ 𝐵)
13		difeq2 4091	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 ∖ 𝑦) → (𝐴 ∖ 𝑥) = (𝐴 ∖ (𝐴 ∖ 𝑦)))
14	13	eleq1d 2895	. . . . . . . 8 ⊢ (𝑥 = (𝐴 ∖ 𝑦) → ((𝐴 ∖ 𝑥) ∈ 𝐵 ↔ (𝐴 ∖ (𝐴 ∖ 𝑦)) ∈ 𝐵))
15	14	rspcev 3621	. . . . . . 7 ⊢ (((𝐴 ∖ 𝑦) ∈ 𝒫 𝐴 ∧ (𝐴 ∖ (𝐴 ∖ 𝑦)) ∈ 𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐴 ∖ 𝑥) ∈ 𝐵)
16	5, 12, 15	syl2anc 586	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐴 ∖ 𝑥) ∈ 𝐵)
17	16	ex 415	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝒫 𝐴(𝐴 ∖ 𝑥) ∈ 𝐵))
18	17	exlimdv 1928	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) → (∃𝑦 𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝒫 𝐴(𝐴 ∖ 𝑥) ∈ 𝐵))
19	1, 18	syl5bi 244	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴) → (𝐵 ≠ ∅ → ∃𝑥 ∈ 𝒫 𝐴(𝐴 ∖ 𝑥) ∈ 𝐵))
20	19	3impia 1112	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴 ∧ 𝐵 ≠ ∅) → ∃𝑥 ∈ 𝒫 𝐴(𝐴 ∖ 𝑥) ∈ 𝐵)
21		rabn0 4337	. 2 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑥) ∈ 𝐵} ≠ ∅ ↔ ∃𝑥 ∈ 𝒫 𝐴(𝐴 ∖ 𝑥) ∈ 𝐵)
22	20, 21	sylibr 236	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝒫 𝐴 ∧ 𝐵 ≠ ∅) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑥) ∈ 𝐵} ≠ ∅)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1082   = wceq 1531  ∃wex 1774   ∈ wcel 2108   ≠ wne 3014  ∃wrex 3137  {crab 3140   ∖ cdif 3931   ⊆ wss 3934  ∅c0 4289  𝒫 cpw 4537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-in 3941  df-ss 3950  df-nul 4290  df-pw 4539

This theorem is referenced by:  fin2i2  9732  isfin2-2  9733