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Theorem rint0 3645
 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 3609 . . 3 (𝑋 = ∅ → 𝑋 = ∅)
21ineq2d 3132 . 2 (𝑋 = ∅ → (A 𝑋) = (A ∅))
3 int0 3620 . . . 4 ∅ = V
43ineq2i 3129 . . 3 (A ∅) = (A ∩ V)
5 inv1 3247 . . 3 (A ∩ V) = A
64, 5eqtri 2057 . 2 (A ∅) = A
72, 6syl6eq 2085 1 (𝑋 = ∅ → (A 𝑋) = A)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242  Vcvv 2551   ∩ cin 2910  ∅c0 3218  ∩ cint 3606 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219  df-int 3607 This theorem is referenced by: (None)
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