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Theorem rint0 4452
 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
StepHypRef Expression
1 inteq 4413 . . 3 (𝑋 = ∅ → 𝑋 = ∅)
21ineq2d 3776 . 2 (𝑋 = ∅ → (𝐴 𝑋) = (𝐴 ∅))
3 int0 4425 . . . 4 ∅ = V
43ineq2i 3773 . . 3 (𝐴 ∅) = (𝐴 ∩ V)
5 inv1 3922 . . 3 (𝐴 ∩ V) = 𝐴
64, 5eqtri 2632 . 2 (𝐴 ∅) = 𝐴
72, 6syl6eq 2660 1 (𝑋 = ∅ → (𝐴 𝑋) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  Vcvv 3173   ∩ cin 3539  ∅c0 3874  ∩ cint 4410 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-int 4411 This theorem is referenced by:  incexclem  14407  incexc  14408  mrerintcl  16080  ismred2  16086  txtube  21253
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