New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  0iin GIF version

Theorem 0iin 4024
 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 3972 . 2 x A = {y x y A}
2 vex 2862 . . . 4 y V
3 ral0 3654 . . . 4 x y A
42, 32th 230 . . 3 (y V ↔ x y A)
54abbi2i 2464 . 2 V = {y x y A}
61, 5eqtr4i 2376 1 x A = V
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614  Vcvv 2859  ∅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-nul 3551  df-iin 3972 This theorem is referenced by:  iinrab2  4029  riin0  4039
 Copyright terms: Public domain W3C validator