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Theorem rint0 4921
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 4882 . . 3 (𝑋 = ∅ → 𝑋 = ∅)
21ineq2d 4146 . 2 (𝑋 = ∅ → (𝐴 𝑋) = (𝐴 ∅))
3 int0 4893 . . . 4 ∅ = V
43ineq2i 4143 . . 3 (𝐴 ∅) = (𝐴 ∩ V)
5 inv1 4328 . . 3 (𝐴 ∩ V) = 𝐴
64, 5eqtri 2766 . 2 (𝐴 ∅) = 𝐴
72, 6eqtrdi 2794 1 (𝑋 = ∅ → (𝐴 𝑋) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  Vcvv 3432  cin 3886  c0 4256   cint 4879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257  df-int 4880
This theorem is referenced by:  incexclem  15548  incexc  15549  mrerintcl  17306  ismred2  17312  txtube  22791  bj-mooreset  35273  bj-ismoored0  35277  bj-ismooredr2  35281
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