Theorem dif1o 6704

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 6694	. . . 4 |- 1o = { (/) }
2	1	difeq2i 3344	. . 3 |- ( B \ 1o ) = ( B \ { (/) } )
3	2	eleq2i 2305	. 2 |- ( A e. ( B \ 1o ) <-> A e. ( B \ { (/) } ) )
4		eldifsn 3839	. 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 2209   =/= wne 2420   \ cdif 3217   (/)c0 3520   {csn 3708   1oc1o 6673

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-nul 3521  df-sn 3714  df-suc 4514  df-1o 6680

This theorem is referenced by: (None)