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Theorem rint0 4652
 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 4615 . . 3 (𝑋 = ∅ → 𝑋 = ∅)
21ineq2d 3965 . 2 (𝑋 = ∅ → (𝐴 𝑋) = (𝐴 ∅))
3 int0 4626 . . . 4 ∅ = V
43ineq2i 3962 . . 3 (𝐴 ∅) = (𝐴 ∩ V)
5 inv1 4115 . . 3 (𝐴 ∩ V) = 𝐴
64, 5eqtri 2793 . 2 (𝐴 ∅) = 𝐴
72, 6syl6eq 2821 1 (𝑋 = ∅ → (𝐴 𝑋) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1631  Vcvv 3351   ∩ cin 3722  ∅c0 4063  ∩ cint 4612 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-v 3353  df-dif 3726  df-in 3730  df-ss 3737  df-nul 4064  df-int 4613 This theorem is referenced by:  incexclem  14775  incexc  14776  mrerintcl  16465  ismred2  16471  txtube  21664  bj-mooreset  33388  bj-ismoored0  33393  bj-ismooredr2  33397
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