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Theorem 0iin 3689
Description: An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
Assertion
Ref Expression
0iin x A = V

Proof of Theorem 0iin
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-iin 3634 . 2 x A = {yx y A}
2 vex 2538 . . . 4 y V
3 ral0 3301 . . . 4 x y A
42, 32th 163 . . 3 (y V ↔ x y A)
54abbi2i 2134 . 2 V = {yx y A}
61, 5eqtr4i 2045 1 x A = V
Colors of variables: wff set class
Syntax hints:   = wceq 1228   wcel 1374  {cab 2008  wral 2284  Vcvv 2535  c0 3201   ciin 3632
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-dif 2897  df-nul 3202  df-iin 3634
This theorem is referenced by:  riin0  3702  iin0r  3896
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