Theorem rint0 4930

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 4892	. . 3 ⊢ (𝑋 = ∅ → ∩ 𝑋 = ∩ ∅)
2	1	ineq2d 4160	. 2 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = (𝐴 ∩ ∩ ∅))
3		int0 4904	. . . 4 ⊢ ∩ ∅ = V
4	3	ineq2i 4157	. . 3 ⊢ (𝐴 ∩ ∩ ∅) = (𝐴 ∩ V)
5		inv1 4338	. . 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 1542  Vcvv 3429   ∩ cin 3888  ∅c0 4273  ∩ cint 4889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-in 3896  df-ss 3906  df-nul 4274  df-int 4890

This theorem is referenced by:  incexclem  15801  incexc  15802  mrerintcl  17559  ismred2  17565  txtube  23605  bj-mooreset  37414  bj-ismoored0  37418  bj-ismooredr2  37422