Theorem rint0 4441

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 4402	. . 3 ⊢ (𝑋 = ∅ → ∩ 𝑋 = ∩ ∅)
2	1	ineq2d 3770	. 2 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = (𝐴 ∩ ∩ ∅))
3		int0 4414	. . . 4 ⊢ ∩ ∅ = V
4	3	ineq2i 3767	. . 3 ⊢ (𝐴 ∩ ∩ ∅) = (𝐴 ∩ V)
5		inv1 3916	. . 3 ⊢ (𝐴 ∩ V) = 𝐴
6	4, 5	eqtri 2626	. 2 ⊢ (𝐴 ∩ ∩ ∅) = 𝐴
7	2, 6	syl6eq 2654	1 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = 𝐴)

Syntax hints:   → wi 4   = wceq 1474  Vcvv 3167   ∩ cin 3533  ∅c0 3868  ∩ cint 4399

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ral 2895  df-v 3169  df-dif 3537  df-in 3541  df-ss 3548  df-nul 3869  df-int 4400

This theorem is referenced by:  incexclem  14348  incexc  14349  mrerintcl  16021  ismred2  16027  txtube  21190