Theorem reupick 4304

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 3952	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
2	1	ad2antrr 726	. 2 ⊢ (((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) ∧ 𝜑) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
3		df-rex 3061	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
4		df-reu 3360	. . . . . 6 ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
5	3, 4	anbi12i 628	. . . . 5 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
6	1	ancrd 551	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)))
7	6	anim1d 611	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑)))
8		an32 646	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 ∈ 𝐴))
9	7, 8	imbitrdi 251	. . . . . . . . 9 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → ((𝑥 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 ∈ 𝐴)))
10	9	eximdv 1917	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 ∈ 𝐴)))
11		eupick 2632	. . . . . . . . 9 ⊢ ((∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑) ∧ ∃𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 ∈ 𝐴)) → ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 ∈ 𝐴))
12	11	ex 412	. . . . . . . 8 ⊢ (∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑) → (∃𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 ∈ 𝐴)))
13	10, 12	syl9 77	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → (∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 ∈ 𝐴))))
14	13	com23 86	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → (∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑) → ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 ∈ 𝐴))))
15	14	imp32 418	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))) → ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 ∈ 𝐴))
16	5, 15	sylan2b 594	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) → ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 ∈ 𝐴))
17	16	expcomd 416	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) → (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴)))
18	17	imp 406	. 2 ⊢ (((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) ∧ 𝜑) → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴))
19	2, 18	impbid 212	1 ⊢ (((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) ∧ 𝜑) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1779   ∈ wcel 2108  ∃!weu 2567  ∃wrex 3060  ∃!wreu 3357   ⊆ wss 3926

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-12 2177

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-mo 2539  df-eu 2568  df-clel 2809  df-rex 3061  df-reu 3360  df-ss 3943

This theorem is referenced by: (None)