Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  acnrcl Unicode version

Theorem acnrcl 7685
 Description: Reverse closure for the choice set predicate. (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
acnrcl AC

Proof of Theorem acnrcl
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ne0i 3474 . . 3
2 abn0 3486 . . . 4
3 simpl 443 . . . . 5
43exlimiv 1624 . . . 4
52, 4sylbi 187 . . 3
61, 5syl 15 . 2
7 df-acn 7591 . 2 AC
86, 7eleq2s 2388 1 AC
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358  wex 1531   wcel 1696  cab 2282   wne 2459  wral 2556  cvv 2801   cdif 3162  c0 3468  cpw 3638  csn 3653  cfv 5271  (class class class)co 5874   cmap 6788  AC wacn 7587 This theorem is referenced by:  acni  7688  acni2  7689  acndom2  7697  fodomacn  7699  iundom2g  8178 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-nul 3469  df-acn 7591
 Copyright terms: Public domain W3C validator