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Theorem rint0 3868
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 3832 . . 3 (𝑋 = ∅ → 𝑋 = ∅)
21ineq2d 3328 . 2 (𝑋 = ∅ → (𝐴 𝑋) = (𝐴 ∅))
3 int0 3843 . . . 4 ∅ = V
43ineq2i 3325 . . 3 (𝐴 ∅) = (𝐴 ∩ V)
5 inv1 3450 . . 3 (𝐴 ∩ V) = 𝐴
64, 5eqtri 2191 . 2 (𝐴 ∅) = 𝐴
72, 6eqtrdi 2219 1 (𝑋 = ∅ → (𝐴 𝑋) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  Vcvv 2730  cin 3120  c0 3414   cint 3829
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-int 3830
This theorem is referenced by: (None)
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