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Theorem riin0 3784
Description: Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
riin0 (𝑋 = ∅ → (𝐴 𝑥𝑋 𝑆) = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑋
Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem riin0
StepHypRef Expression
1 iineq1 3727 . . 3 (𝑋 = ∅ → 𝑥𝑋 𝑆 = 𝑥 ∈ ∅ 𝑆)
21ineq2d 3190 . 2 (𝑋 = ∅ → (𝐴 𝑥𝑋 𝑆) = (𝐴 𝑥 ∈ ∅ 𝑆))
3 0iin 3771 . . . 4 𝑥 ∈ ∅ 𝑆 = V
43ineq2i 3187 . . 3 (𝐴 𝑥 ∈ ∅ 𝑆) = (𝐴 ∩ V)
5 inv1 3307 . . 3 (𝐴 ∩ V) = 𝐴
64, 5eqtri 2105 . 2 (𝐴 𝑥 ∈ ∅ 𝑆) = 𝐴
72, 6syl6eq 2133 1 (𝑋 = ∅ → (𝐴 𝑥𝑋 𝑆) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  Vcvv 2615  cin 2987  c0 3275   ciin 3714
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2617  df-dif 2990  df-in 2994  df-ss 3001  df-nul 3276  df-iin 3716
This theorem is referenced by: (None)
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