Theorem dif1o 7147

Description: Two ways to say that A is a nonzero number of the set B. (Contributed by Mario Carneiro, 21-May-2015.)

Assertion

Ref	Expression
dif1o	|- ( A e. ( B \ 1o ) <-> ( A e. B /\ A =/= (/) ) )

Proof of Theorem dif1o

Step	Hyp	Ref	Expression
1		df1o2 7139	. . . 4 |- 1o = { (/) }
2	1	difeq2i 3619	. . 3 |- ( B \ 1o ) = ( B \ { (/) } )
3	2	eleq2i 2545	. 2 |- ( A e. ( B \ 1o ) <-> A e. ( B \ { (/) } ) )
4		eldifsn 4152	. 2 |- ( A e. ( B \ { (/) } ) <-> ( A e. B /\ A =/= (/) ) )
5	3, 4	bitri 249	1 |- ( A e. ( B \ 1o ) <-> ( A e. B /\ A =/= (/) ) )

Syntax hints:   <-> wb 184   /\ wa 369   e. wcel 1767   =/= wne 2662   \ cdif 3473   (/)c0 3785   {csn 4027   1oc1o 7120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-nul 3786  df-sn 4028  df-suc 4884  df-1o 7127

This theorem is referenced by:  ondif1  7148  brwitnlem  7154  oelim2  7241  oeeulem  7247  oeeui  7248  omabs  7293  cantnfp1lem3  8095  cantnfp1  8096  cantnflem1  8104  cantnflem3  8106  cantnflem4  8107  cantnfleOLD  8116  cantnfp1lem2OLD  8120  cantnfp1lem3OLD  8121  cantnfp1OLD  8122  cantnflem1aOLD  8123  cantnflem1cOLD  8125  cantnflem1OLD  8127  cantnflem3OLD  8128  cantnflem4OLD  8129  cantnfOLD  8130  cnfcom3lem  8143  cnfcomlemOLD  8147  cnfcom3lemOLD  8151  cnfcom3OLD  8152