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Theorem riin0 3852
 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 3795 . . 3 (𝑋 = ∅ → 𝑥𝑋 𝑆 = 𝑥 ∈ ∅ 𝑆)
21ineq2d 3245 . 2 (𝑋 = ∅ → (𝐴 𝑥𝑋 𝑆) = (𝐴 𝑥 ∈ ∅ 𝑆))
3 0iin 3839 . . . 4 𝑥 ∈ ∅ 𝑆 = V
43ineq2i 3242 . . 3 (𝐴 𝑥 ∈ ∅ 𝑆) = (𝐴 ∩ V)
5 inv1 3367 . . 3 (𝐴 ∩ V) = 𝐴
64, 5eqtri 2136 . 2 (𝐴 𝑥 ∈ ∅ 𝑆) = 𝐴
72, 6syl6eq 2164 1 (𝑋 = ∅ → (𝐴 𝑥𝑋 𝑆) = 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1314  Vcvv 2658   ∩ cin 3038  ∅c0 3331  ∩ ciin 3782 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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097 This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660  df-dif 3041  df-in 3045  df-ss 3052  df-nul 3332  df-iin 3784 This theorem is referenced by: (None)
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