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Theorem rint0 4549
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 4510 . . 3 (𝑋 = ∅ → 𝑋 = ∅)
21ineq2d 3847 . 2 (𝑋 = ∅ → (𝐴 𝑋) = (𝐴 ∅))
3 int0 4522 . . . 4 ∅ = V
43ineq2i 3844 . . 3 (𝐴 ∅) = (𝐴 ∩ V)
5 inv1 4003 . . 3 (𝐴 ∩ V) = 𝐴
64, 5eqtri 2673 . 2 (𝐴 ∅) = 𝐴
72, 6syl6eq 2701 1 (𝑋 = ∅ → (𝐴 𝑋) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  Vcvv 3231  cin 3606  c0 3948   cint 4507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-int 4508
This theorem is referenced by:  incexclem  14612  incexc  14613  mrerintcl  16304  ismred2  16310  txtube  21491  bj-mooreset  33181  bj-ismoored0  33186  bj-ismooredr2  33190
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