Theorem riinn0 4040

Description: Relative intersection of a nonempty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Assertion

Ref	Expression
riinn0	⊢ ((∀x ∈ X S ⊆ A ∧ X ≠ ∅) → (A ∩ ∩x ∈ X S) = ∩x ∈ X S)

Distinct variable groups:   x,A   x,X

Allowed substitution hint:   S(x)

Proof of Theorem riinn0

Step	Hyp	Ref	Expression
1		incom 3448	. 2 ⊢ (A ∩ ∩x ∈ X S) = (∩x ∈ X S ∩ A)
2		r19.2z 3639	. . . . 5 ⊢ ((X ≠ ∅ ∧ ∀x ∈ X S ⊆ A) → ∃x ∈ X S ⊆ A)
3	2	ancoms 439	. . . 4 ⊢ ((∀x ∈ X S ⊆ A ∧ X ≠ ∅) → ∃x ∈ X S ⊆ A)
4		iinss 4017	. . . 4 ⊢ (∃x ∈ X S ⊆ A → ∩x ∈ X S ⊆ A)
5	3, 4	syl 15	. . 3 ⊢ ((∀x ∈ X S ⊆ A ∧ X ≠ ∅) → ∩x ∈ X S ⊆ A)
6		df-ss 3259	. . 3 ⊢ (∩x ∈ X S ⊆ A ↔ (∩x ∈ X S ∩ A) = ∩x ∈ X S)
7	5, 6	sylib 188	. 2 ⊢ ((∀x ∈ X S ⊆ A ∧ X ≠ ∅) → (∩x ∈ X S ∩ A) = ∩x ∈ X S)
8	1, 7	syl5eq 2397	1 ⊢ ((∀x ∈ X S ⊆ A ∧ X ≠ ∅) → (A ∩ ∩x ∈ X S) = ∩x ∈ X S)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ≠ wne 2516  ∀wral 2614  ∃wrex 2615   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550  ∩ciin 3970

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551  df-iin 3972

This theorem is referenced by:  riinrab  4041