Theorem rint0 4943

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

Assertion

Ref	Expression
rint0	⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = 𝐴)

Proof of Theorem rint0

Step	Hyp	Ref	Expression
1		inteq 4905	. . 3 ⊢ (𝑋 = ∅ → ∩ 𝑋 = ∩ ∅)
2	1	ineq2d 4172	. 2 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = (𝐴 ∩ ∩ ∅))
3		int0 4917	. . . 4 ⊢ ∩ ∅ = V
4	3	ineq2i 4169	. . 3 ⊢ (𝐴 ∩ ∩ ∅) = (𝐴 ∩ V)
5		inv1 4350	. . 3 ⊢ (𝐴 ∩ V) = 𝐴
6	4, 5	eqtri 2759	. 2 ⊢ (𝐴 ∩ ∩ ∅) = 𝐴
7	2, 6	eqtrdi 2787	1 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = 𝐴)

Syntax hints:   → wi 4   = wceq 1541  Vcvv 3440   ∩ cin 3900  ∅c0 4285  ∩ cint 4902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-in 3908  df-ss 3918  df-nul 4286  df-int 4903

This theorem is referenced by:  incexclem  15759  incexc  15760  mrerintcl  17516  ismred2  17522  txtube  23584  bj-mooreset  37303  bj-ismoored0  37307  bj-ismooredr2  37311