Theorem dif1o 6496

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 6487	. . . 4 |- 1o = { (/) }
2	1	difeq2i 3278	. . 3 |- ( B \ 1o ) = ( B \ { (/) } )
3	2	eleq2i 2263	. 2 |- ( A e. ( B \ 1o ) <-> A e. ( B \ { (/) } ) )
4		eldifsn 3749	. 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 2167   =/= wne 2367   \ cdif 3154   (/)c0 3450   {csn 3622   1oc1o 6467

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-nul 3451  df-sn 3628  df-suc 4406  df-1o 6474

This theorem is referenced by: (None)