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Theorem rint0 4994
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 4953 . . 3 (𝑋 = ∅ → 𝑋 = ∅)
21ineq2d 4212 . 2 (𝑋 = ∅ → (𝐴 𝑋) = (𝐴 ∅))
3 int0 4966 . . . 4 ∅ = V
43ineq2i 4209 . . 3 (𝐴 ∅) = (𝐴 ∩ V)
5 inv1 4394 . . 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 3475  cin 3947  c0 4322   cint 4950
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-in 3955  df-ss 3965  df-nul 4323  df-int 4951
This theorem is referenced by:  incexclem  15779  incexc  15780  mrerintcl  17538  ismred2  17544  txtube  23136  bj-mooreset  35972  bj-ismoored0  35976  bj-ismooredr2  35980
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