Theorem reupick 3447

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 3177	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
2	1	ad2antrr 488	. 2 ⊢ (((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) ∧ 𝜑) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
3		df-rex 2481	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
4		df-reu 2482	. . . . . 6 ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
5	3, 4	anbi12i 460	. . . . 5 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
6	1	ancrd 326	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)))
7	6	anim1d 336	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑)))
8		an32 562	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 ∈ 𝐴))
9	7, 8	imbitrdi 161	. . . . . . . . 9 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → ((𝑥 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 ∈ 𝐴)))
10	9	eximdv 1894	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 ∈ 𝐴)))
11		eupick 2124	. . . . . . . . 9 ⊢ ((∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑) ∧ ∃𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 ∈ 𝐴)) → ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 ∈ 𝐴))
12	11	ex 115	. . . . . . . 8 ⊢ (∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑) → (∃𝑥((𝑥 ∈ 𝐵 ∧ 𝜑) ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 ∈ 𝐴)))
13	10, 12	syl9 72	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → (∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 ∈ 𝐴))))
14	13	com23 78	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → (∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑) → ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 ∈ 𝐴))))
15	14	imp32 257	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))) → ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 ∈ 𝐴))
16	5, 15	sylan2b 287	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) → ((𝑥 ∈ 𝐵 ∧ 𝜑) → 𝑥 ∈ 𝐴))
17	16	expcomd 1452	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) → (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴)))
18	17	imp 124	. 2 ⊢ (((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) ∧ 𝜑) → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴))
19	2, 18	impbid 129	1 ⊢ (((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) ∧ 𝜑) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∃wex 1506  ∃!weu 2045   ∈ wcel 2167  ∃wrex 2476  ∃!wreu 2477   ⊆ wss 3157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-rex 2481  df-reu 2482  df-in 3163  df-ss 3170

This theorem is referenced by:  supelti  7068