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Theorem rint0 3810
 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 3774 . . 3 (𝑋 = ∅ → 𝑋 = ∅)
21ineq2d 3277 . 2 (𝑋 = ∅ → (𝐴 𝑋) = (𝐴 ∅))
3 int0 3785 . . . 4 ∅ = V
43ineq2i 3274 . . 3 (𝐴 ∅) = (𝐴 ∩ V)
5 inv1 3399 . . 3 (𝐴 ∩ V) = 𝐴
64, 5eqtri 2160 . 2 (𝐴 ∅) = 𝐴
72, 6syl6eq 2188 1 (𝑋 = ∅ → (𝐴 𝑋) = 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331  Vcvv 2686   ∩ cin 3070  ∅c0 3363  ∩ cint 3771 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-nul 3364  df-int 3772 This theorem is referenced by: (None)
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