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Theorem rint0 3898
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 3862 . . 3 (𝑋 = ∅ → 𝑋 = ∅)
21ineq2d 3351 . 2 (𝑋 = ∅ → (𝐴 𝑋) = (𝐴 ∅))
3 int0 3873 . . . 4 ∅ = V
43ineq2i 3348 . . 3 (𝐴 ∅) = (𝐴 ∩ V)
5 inv1 3474 . . 3 (𝐴 ∩ V) = 𝐴
64, 5eqtri 2210 . 2 (𝐴 ∅) = 𝐴
72, 6eqtrdi 2238 1 (𝑋 = ∅ → (𝐴 𝑋) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  Vcvv 2752  cin 3143  c0 3437   cint 3859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-nul 3438  df-int 3860
This theorem is referenced by: (None)
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