Theorem dif1o 6584

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 6575	. . . 4 |- 1o = { (/) }
2	1	difeq2i 3319	. . 3 |- ( B \ 1o ) = ( B \ { (/) } )
3	2	eleq2i 2296	. 2 |- ( A e. ( B \ 1o ) <-> A e. ( B \ { (/) } ) )
4		eldifsn 3795	. 2 |- ( A e. ( B \ { (/) } ) <-> ( A e. B /\ A =/= (/) ) )
5	3, 4	bitri 184	1 |- ( A e. ( B \ 1o ) <-> ( A e. B /\ A =/= (/) ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2200   =/= wne 2400   \ cdif 3194   (/)c0 3491   {csn 3666   1oc1o 6555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-nul 3492  df-sn 3672  df-suc 4462  df-1o 6562

This theorem is referenced by: (None)