Theorem iinss 4018

Description: Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iinss	⊢ (∃x ∈ A B ⊆ C → ∩x ∈ A B ⊆ C)

Distinct variable group:   x,C

Allowed substitution hints:   A(x)   B(x)

Proof of Theorem iinss

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2863	. . . 4 ⊢ y ∈ V
2		eliin 3975	. . . 4 ⊢ (y ∈ V → (y ∈ ∩x ∈ A B ↔ ∀x ∈ A y ∈ B))
3	1, 2	ax-mp 5	. . 3 ⊢ (y ∈ ∩x ∈ A B ↔ ∀x ∈ A y ∈ B)
4		ssel 3268	. . . . 5 ⊢ (B ⊆ C → (y ∈ B → y ∈ C))
5	4	reximi 2722	. . . 4 ⊢ (∃x ∈ A B ⊆ C → ∃x ∈ A (y ∈ B → y ∈ C))
6		r19.36av 2760	. . . 4 ⊢ (∃x ∈ A (y ∈ B → y ∈ C) → (∀x ∈ A y ∈ B → y ∈ C))
7	5, 6	syl 15	. . 3 ⊢ (∃x ∈ A B ⊆ C → (∀x ∈ A y ∈ B → y ∈ C))
8	3, 7	syl5bi 208	. 2 ⊢ (∃x ∈ A B ⊆ C → (y ∈ ∩x ∈ A B → y ∈ C))
9	8	ssrdv 3279	1 ⊢ (∃x ∈ A B ⊆ C → ∩x ∈ A B ⊆ C)

Syntax hints:   → wi 4   ↔ wb 176   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616  Vcvv 2860   ⊆ wss 3258  ∩ciin 3971

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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-iin 3973

This theorem is referenced by:  riinn0  4041