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Theorem rint0 4441
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 4402 . . 3 (𝑋 = ∅ → 𝑋 = ∅)
21ineq2d 3770 . 2 (𝑋 = ∅ → (𝐴 𝑋) = (𝐴 ∅))
3 int0 4414 . . . 4 ∅ = V
43ineq2i 3767 . . 3 (𝐴 ∅) = (𝐴 ∩ V)
5 inv1 3916 . . 3 (𝐴 ∩ V) = 𝐴
64, 5eqtri 2626 . 2 (𝐴 ∅) = 𝐴
72, 6syl6eq 2654 1 (𝑋 = ∅ → (𝐴 𝑋) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  Vcvv 3167  cin 3533  c0 3868   cint 4399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ral 2895  df-v 3169  df-dif 3537  df-in 3541  df-ss 3548  df-nul 3869  df-int 4400
This theorem is referenced by:  incexclem  14348  incexc  14349  mrerintcl  16021  ismred2  16027  txtube  21190
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