Theorem reupick 4205

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 3868	. . 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 3059	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
4		df-reu 3060	. . . . . 6 ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
5	3, 4	anbi12i 630	. . . . 5 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
6	1	ancrd 555	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)))
7	6	anim1d 614	. . . . . . . . . 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	syl6ib 254	. . . . . . . . 9 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → ((𝑥 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 ∈ 𝐴)))
10	9	eximdv 1923	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 ∈ 𝐴)))
11		eupick 2636	. . . . . . . . 9 ⊢ ((∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑) ∧ ∃𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 ∈ 𝐴)) → ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 ∈ 𝐴))
12	11	ex 416	. . . . . . . 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 422	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))) → ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 ∈ 𝐴))
16	5, 15	sylan2b 597	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) → ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 ∈ 𝐴))
17	16	expcomd 420	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) → (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴)))
18	17	imp 410	. 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))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∃wex 1786   ∈ wcel 2113  ∃!weu 2569  ∃wrex 3054  ∃!wreu 3055   ⊆ wss 3841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-12 2178  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-rex 3059  df-reu 3060  df-v 3399  df-in 3848  df-ss 3858

This theorem is referenced by: (None)