Theorem rint0 4993

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 4954	. . 3 ⊢ (𝑋 = ∅ → ∩ 𝑋 = ∩ ∅)
2	1	ineq2d 4228	. 2 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = (𝐴 ∩ ∩ ∅))
3		int0 4967	. . . 4 ⊢ ∩ ∅ = V
4	3	ineq2i 4225	. . 3 ⊢ (𝐴 ∩ ∩ ∅) = (𝐴 ∩ V)
5		inv1 4404	. . 3 ⊢ (𝐴 ∩ V) = 𝐴
6	4, 5	eqtri 2763	. 2 ⊢ (𝐴 ∩ ∩ ∅) = 𝐴
7	2, 6	eqtrdi 2791	1 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = 𝐴)

Syntax hints:   → wi 4   = wceq 1537  Vcvv 3478   ∩ cin 3962  ∅c0 4339  ∩ cint 4951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340  df-int 4952

This theorem is referenced by:  incexclem  15869  incexc  15870  mrerintcl  17642  ismred2  17648  txtube  23664  bj-mooreset  37085  bj-ismoored0  37089  bj-ismooredr2  37093