Theorem reupick 3540

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 3268	. . 3 ⊢ (A ⊆ B → (x ∈ A → x ∈ B))
2	1	ad2antrr 706	. 2 ⊢ (((A ⊆ B ∧ (∃x ∈ A φ ∧ ∃!x ∈ B φ)) ∧ φ) → (x ∈ A → x ∈ B))
3		df-rex 2621	. . . . . 6 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
4		df-reu 2622	. . . . . 6 ⊢ (∃!x ∈ B φ ↔ ∃!x(x ∈ B ∧ φ))
5	3, 4	anbi12i 678	. . . . 5 ⊢ ((∃x ∈ A φ ∧ ∃!x ∈ B φ) ↔ (∃x(x ∈ A ∧ φ) ∧ ∃!x(x ∈ B ∧ φ)))
6	1	ancrd 537	. . . . . . . . . . 11 ⊢ (A ⊆ B → (x ∈ A → (x ∈ B ∧ x ∈ A)))
7	6	anim1d 547	. . . . . . . . . 10 ⊢ (A ⊆ B → ((x ∈ A ∧ φ) → ((x ∈ B ∧ x ∈ A) ∧ φ)))
8		an32 773	. . . . . . . . . 10 ⊢ (((x ∈ B ∧ x ∈ A) ∧ φ) ↔ ((x ∈ B ∧ φ) ∧ x ∈ A))
9	7, 8	syl6ib 217	. . . . . . . . 9 ⊢ (A ⊆ B → ((x ∈ A ∧ φ) → ((x ∈ B ∧ φ) ∧ x ∈ A)))
10	9	eximdv 1622	. . . . . . . 8 ⊢ (A ⊆ B → (∃x(x ∈ A ∧ φ) → ∃x((x ∈ B ∧ φ) ∧ x ∈ A)))
11		eupick 2267	. . . . . . . . 9 ⊢ ((∃!x(x ∈ B ∧ φ) ∧ ∃x((x ∈ B ∧ φ) ∧ x ∈ A)) → ((x ∈ B ∧ φ) → x ∈ A))
12	11	ex 423	. . . . . . . 8 ⊢ (∃!x(x ∈ B ∧ φ) → (∃x((x ∈ B ∧ φ) ∧ x ∈ A) → ((x ∈ B ∧ φ) → x ∈ A)))
13	10, 12	syl9 66	. . . . . . 7 ⊢ (A ⊆ B → (∃!x(x ∈ B ∧ φ) → (∃x(x ∈ A ∧ φ) → ((x ∈ B ∧ φ) → x ∈ A))))
14	13	com23 72	. . . . . 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 461	. . . 4 ⊢ ((A ⊆ B ∧ (∃x ∈ A φ ∧ ∃!x ∈ B φ)) → ((x ∈ B ∧ φ) → x ∈ A))
17	16	exp3acom23 1372	. . 3 ⊢ ((A ⊆ B ∧ (∃x ∈ A φ ∧ ∃!x ∈ B φ)) → (φ → (x ∈ B → x ∈ A)))
18	17	imp 418	. 2 ⊢ (((A ⊆ B ∧ (∃x ∈ A φ ∧ ∃!x ∈ B φ)) ∧ φ) → (x ∈ B → x ∈ A))
19	2, 18	impbid 183	1 ⊢ (((A ⊆ B ∧ (∃x ∈ A φ ∧ ∃!x ∈ B φ)) ∧ φ) → (x ∈ A ↔ x ∈ B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  ∃!weu 2204  ∃wrex 2616  ∃!wreu 2617   ⊆ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-reu 2622  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by: (None)