Theorem rint0 5012

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 4973	. . 3 ⊢ (𝑋 = ∅ → ∩ 𝑋 = ∩ ∅)
2	1	ineq2d 4241	. 2 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = (𝐴 ∩ ∩ ∅))
3		int0 4986	. . . 4 ⊢ ∩ ∅ = V
4	3	ineq2i 4238	. . 3 ⊢ (𝐴 ∩ ∩ ∅) = (𝐴 ∩ V)
5		inv1 4421	. . 3 ⊢ (𝐴 ∩ V) = 𝐴
6	4, 5	eqtri 2768	. 2 ⊢ (𝐴 ∩ ∩ ∅) = 𝐴
7	2, 6	eqtrdi 2796	1 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = 𝐴)

Syntax hints:   → wi 4   = wceq 1537  Vcvv 3488   ∩ cin 3975  ∅c0 4352  ∩ cint 4970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-in 3983  df-ss 3993  df-nul 4353  df-int 4971

This theorem is referenced by:  incexclem  15884  incexc  15885  mrerintcl  17655  ismred2  17661  txtube  23669  bj-mooreset  37068  bj-ismoored0  37072  bj-ismooredr2  37076