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Theorem rint0 3695
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 3659 . . 3 (𝑋 = ∅ → 𝑋 = ∅)
21ineq2d 3183 . 2 (𝑋 = ∅ → (𝐴 𝑋) = (𝐴 ∅))
3 int0 3670 . . . 4 ∅ = V
43ineq2i 3180 . . 3 (𝐴 ∅) = (𝐴 ∩ V)
5 inv1 3296 . . 3 (𝐴 ∩ V) = 𝐴
64, 5eqtri 2103 . 2 (𝐴 ∅) = 𝐴
72, 6syl6eq 2131 1 (𝑋 = ∅ → (𝐴 𝑋) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  Vcvv 2610  cin 2981  c0 3267   cint 3656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2612  df-dif 2984  df-in 2988  df-ss 2995  df-nul 3268  df-int 3657
This theorem is referenced by: (None)
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