Theorem riin0 4040

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

Assertion

Ref	Expression
riin0	⊢ (X = ∅ → (A ∩ ∩x ∈ X S) = A)

Distinct variable groups:   x,A   x,X

Allowed substitution hint:   S(x)

Proof of Theorem riin0

Step	Hyp	Ref	Expression
1		iineq1 3984	. . 3 ⊢ (X = ∅ → ∩x ∈ X S = ∩x ∈ ∅ S)
2	1	ineq2d 3458	. 2 ⊢ (X = ∅ → (A ∩ ∩x ∈ X S) = (A ∩ ∩x ∈ ∅ S))
3		0iin 4025	. . . 4 ⊢ ∩x ∈ ∅ S = V
4	3	ineq2i 3455	. . 3 ⊢ (A ∩ ∩x ∈ ∅ S) = (A ∩ V)
5		inv1 3578	. . 3 ⊢ (A ∩ V) = A
6	4, 5	eqtri 2373	. 2 ⊢ (A ∩ ∩x ∈ ∅ S) = A
7	2, 6	syl6eq 2401	1 ⊢ (X = ∅ → (A ∩ ∩x ∈ X S) = A)

Syntax hints:   → wi 4   = wceq 1642  Vcvv 2860   ∩ cin 3209  ∅c0 3551  ∩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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552  df-iin 3973

This theorem is referenced by:  riinrab  4042