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Theorem rint0 4897
 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 4860 . . 3 (𝑋 = ∅ → 𝑋 = ∅)
21ineq2d 4172 . 2 (𝑋 = ∅ → (𝐴 𝑋) = (𝐴 ∅))
3 int0 4871 . . . 4 ∅ = V
43ineq2i 4169 . . 3 (𝐴 ∅) = (𝐴 ∩ V)
5 inv1 4329 . . 3 (𝐴 ∩ V) = 𝐴
64, 5eqtri 2847 . 2 (𝐴 ∅) = 𝐴
72, 6syl6eq 2875 1 (𝑋 = ∅ → (𝐴 𝑋) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  Vcvv 3479   ∩ cin 3917  ∅c0 4274  ∩ cint 4857 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rab 3141  df-v 3481  df-dif 3921  df-in 3925  df-ss 3935  df-nul 4275  df-int 4858 This theorem is referenced by:  incexclem  15180  incexc  15181  mrerintcl  16857  ismred2  16863  txtube  22234  bj-mooreset  34422  bj-ismoored0  34426  bj-ismooredr2  34430
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