Theorem dif1o 6673

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 6663	. . . 4 |- 1o = { (/) }
2	1	difeq2i 3336	. . 3 |- ( B \ 1o ) = ( B \ { (/) } )
3	2	eleq2i 2301	. 2 |- ( A e. ( B \ 1o ) <-> A e. ( B \ { (/) } ) )
4		eldifsn 3822	. 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 2205   =/= wne 2414   \ cdif 3210   (/)c0 3510   {csn 3691   1oc1o 6642

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rab 2531  df-v 2817  df-dif 3215  df-un 3217  df-nul 3511  df-sn 3697  df-suc 4494  df-1o 6649

This theorem is referenced by: (None)