Theorem rint0 4909

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 4872	. . 3 ⊢ (𝑋 = ∅ → ∩ 𝑋 = ∩ ∅)
2	1	ineq2d 4189	. 2 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = (𝐴 ∩ ∩ ∅))
3		int0 4883	. . . 4 ⊢ ∩ ∅ = V
4	3	ineq2i 4186	. . 3 ⊢ (𝐴 ∩ ∩ ∅) = (𝐴 ∩ V)
5		inv1 4348	. . 3 ⊢ (𝐴 ∩ V) = 𝐴
6	4, 5	eqtri 2844	. 2 ⊢ (𝐴 ∩ ∩ ∅) = 𝐴
7	2, 6	syl6eq 2872	1 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = 𝐴)

Syntax hints:   → wi 4   = wceq 1533  Vcvv 3495   ∩ cin 3935  ∅c0 4291  ∩ cint 4869

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3497  df-dif 3939  df-in 3943  df-ss 3952  df-nul 4292  df-int 4870

This theorem is referenced by:  incexclem  15185  incexc  15186  mrerintcl  16862  ismred2  16868  txtube  22242  bj-mooreset  34388  bj-ismoored0  34392  bj-ismooredr2  34396