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Theorem rint0 3735
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 3699 . . 3 (𝑋 = ∅ → 𝑋 = ∅)
21ineq2d 3204 . 2 (𝑋 = ∅ → (𝐴 𝑋) = (𝐴 ∅))
3 int0 3710 . . . 4 ∅ = V
43ineq2i 3201 . . 3 (𝐴 ∅) = (𝐴 ∩ V)
5 inv1 3325 . . 3 (𝐴 ∩ V) = 𝐴
64, 5eqtri 2109 . 2 (𝐴 ∅) = 𝐴
72, 6syl6eq 2137 1 (𝑋 = ∅ → (𝐴 𝑋) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1290  Vcvv 2622  cin 3001  c0 3289   cint 3696
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-v 2624  df-dif 3004  df-in 3008  df-ss 3015  df-nul 3290  df-int 3697
This theorem is referenced by: (None)
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