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Theorem 0iin 3706
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 3651 . 2 x A = {yx y A}
2 vex 2554 . . . 4 y V
3 ral0 3316 . . . 4 x y A
42, 32th 163 . . 3 (y V ↔ x y A)
54abbi2i 2149 . 2 V = {yx y A}
61, 5eqtr4i 2060 1 x A = V
Colors of variables: wff set class
Syntax hints:   = wceq 1242   wcel 1390  {cab 2023  wral 2300  Vcvv 2551  c0 3218   ciin 3649
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-nul 3219  df-iin 3651
This theorem is referenced by:  riin0  3719  iin0r  3913
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