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Theorem rint0 4878
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 4839 . . 3 (𝑋 = ∅ → 𝑋 = ∅)
21ineq2d 4103 . 2 (𝑋 = ∅ → (𝐴 𝑋) = (𝐴 ∅))
3 int0 4850 . . . 4 ∅ = V
43ineq2i 4100 . . 3 (𝐴 ∅) = (𝐴 ∩ V)
5 inv1 4283 . . 3 (𝐴 ∩ V) = 𝐴
64, 5eqtri 2761 . 2 (𝐴 ∅) = 𝐴
72, 6eqtrdi 2789 1 (𝑋 = ∅ → (𝐴 𝑋) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  Vcvv 3398  cin 3842  c0 4211   cint 4836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rab 3062  df-v 3400  df-dif 3846  df-in 3850  df-ss 3860  df-nul 4212  df-int 4837
This theorem is referenced by:  incexclem  15286  incexc  15287  mrerintcl  16973  ismred2  16979  txtube  22393  bj-mooreset  34916  bj-ismoored0  34920  bj-ismooredr2  34924
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