Theorem rint0 4549

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 4510	. . 3 ⊢ (𝑋 = ∅ → ∩ 𝑋 = ∩ ∅)
2	1	ineq2d 3847	. 2 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = (𝐴 ∩ ∩ ∅))
3		int0 4522	. . . 4 ⊢ ∩ ∅ = V
4	3	ineq2i 3844	. . 3 ⊢ (𝐴 ∩ ∩ ∅) = (𝐴 ∩ V)
5		inv1 4003	. . 3 ⊢ (𝐴 ∩ V) = 𝐴
6	4, 5	eqtri 2673	. 2 ⊢ (𝐴 ∩ ∩ ∅) = 𝐴
7	2, 6	syl6eq 2701	1 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = 𝐴)

Syntax hints:   → wi 4   = wceq 1523  Vcvv 3231   ∩ cin 3606  ∅c0 3948  ∩ cint 4507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-int 4508

This theorem is referenced by:  incexclem  14612  incexc  14613  mrerintcl  16304  ismred2  16310  txtube  21491  bj-mooreset  33181  bj-ismoored0  33186  bj-ismooredr2  33190