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Theorem rint0 4988
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 4949 . . 3 (𝑋 = ∅ → 𝑋 = ∅)
21ineq2d 4220 . 2 (𝑋 = ∅ → (𝐴 𝑋) = (𝐴 ∅))
3 int0 4962 . . . 4 ∅ = V
43ineq2i 4217 . . 3 (𝐴 ∅) = (𝐴 ∩ V)
5 inv1 4398 . . 3 (𝐴 ∩ V) = 𝐴
64, 5eqtri 2765 . 2 (𝐴 ∅) = 𝐴
72, 6eqtrdi 2793 1 (𝑋 = ∅ → (𝐴 𝑋) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  Vcvv 3480  cin 3950  c0 4333   cint 4946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-in 3958  df-ss 3968  df-nul 4334  df-int 4947
This theorem is referenced by:  incexclem  15872  incexc  15873  mrerintcl  17640  ismred2  17646  txtube  23648  bj-mooreset  37103  bj-ismoored0  37107  bj-ismooredr2  37111
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