Theorem reupick 4282

Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)

Assertion

Ref	Expression
reupick	⊢ (((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) ∧ 𝜑) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem reupick

Step	Hyp	Ref	Expression
1		ssel 3932	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
2	1	ad2antrr 739	. 2 ⊢ (((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) ∧ 𝜑) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
3		df-rex 3092	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
4		df-reu 3372	. . . . . 6 ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
5	3, 4	anbi12i 640	. . . . 5 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
6	1	ancrd 561	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)))
7	6	anim1d 623	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑)))
8		an32 659	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 ∈ 𝐴))
9	7, 8	imbitrdi 254	. . . . . . . . 9 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → ((𝑥 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 ∈ 𝐴)))
10	9	eximdv 1950	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 ∈ 𝐴)))
11		eupick 2663	. . . . . . . . 9 ⊢ ((∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑) ∧ ∃𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 ∈ 𝐴)) → ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 ∈ 𝐴))
12	11	ex 418	. . . . . . . 8 ⊢ (∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑) → (∃𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 ∈ 𝐴)))
13	10, 12	syl9 78	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → (∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 ∈ 𝐴))))
14	13	com23 87	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → (∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑) → ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 ∈ 𝐴))))
15	14	imp32 424	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))) → ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 ∈ 𝐴))
16	5, 15	sylan2b 606	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) → ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 ∈ 𝐴))
17	16	expcomd 422	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) → (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴)))
18	17	imp 412	. 2 ⊢ (((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) ∧ 𝜑) → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴))
19	2, 18	impbid 215	1 ⊢ (((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) ∧ 𝜑) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∃wex 1812   ∈ wcel 2146  ∃!weu 2598  ∃wrex 3091  ∃!wreu 3369   ⊆ wss 3906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2569  df-eu 2599  df-clel 2840  df-rex 3092  df-reu 3372  df-ss 3923

This theorem is used by: (None)