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Theorem riin0 4039
 Description: Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
riin0 (X = → (Ax X S) = A)
Distinct variable groups:   x,A   x,X
Allowed substitution hint:   S(x)

Proof of Theorem riin0
StepHypRef Expression
1 iineq1 3983 . . 3 (X = x X S = x S)
21ineq2d 3457 . 2 (X = → (Ax X S) = (Ax S))
3 0iin 4024 . . . 4 x S = V
43ineq2i 3454 . . 3 (Ax S) = (A ∩ V)
5 inv1 3577 . . 3 (A ∩ V) = A
64, 5eqtri 2373 . 2 (Ax S) = A
72, 6syl6eq 2401 1 (X = → (Ax X S) = A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642  Vcvv 2859   ∩ cin 3208  ∅c0 3550  ∩ciin 3970 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551  df-iin 3972 This theorem is referenced by:  riinrab  4041
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