Step |
Hyp |
Ref |
Expression |
1 |
|
isnzr2.b |
|- B = ( Base ` R ) |
2 |
|
eqid |
|- ( 1r ` R ) = ( 1r ` R ) |
3 |
|
eqid |
|- ( 0g ` R ) = ( 0g ` R ) |
4 |
2 3
|
isnzr |
|- ( R e. NzRing <-> ( R e. Ring /\ ( 1r ` R ) =/= ( 0g ` R ) ) ) |
5 |
1 2
|
ringidcl |
|- ( R e. Ring -> ( 1r ` R ) e. B ) |
6 |
5
|
adantr |
|- ( ( R e. Ring /\ ( 1r ` R ) =/= ( 0g ` R ) ) -> ( 1r ` R ) e. B ) |
7 |
1 3
|
ring0cl |
|- ( R e. Ring -> ( 0g ` R ) e. B ) |
8 |
7
|
adantr |
|- ( ( R e. Ring /\ ( 1r ` R ) =/= ( 0g ` R ) ) -> ( 0g ` R ) e. B ) |
9 |
|
simpr |
|- ( ( R e. Ring /\ ( 1r ` R ) =/= ( 0g ` R ) ) -> ( 1r ` R ) =/= ( 0g ` R ) ) |
10 |
|
df-ne |
|- ( x =/= y <-> -. x = y ) |
11 |
|
neeq1 |
|- ( x = ( 1r ` R ) -> ( x =/= y <-> ( 1r ` R ) =/= y ) ) |
12 |
10 11
|
bitr3id |
|- ( x = ( 1r ` R ) -> ( -. x = y <-> ( 1r ` R ) =/= y ) ) |
13 |
|
neeq2 |
|- ( y = ( 0g ` R ) -> ( ( 1r ` R ) =/= y <-> ( 1r ` R ) =/= ( 0g ` R ) ) ) |
14 |
12 13
|
rspc2ev |
|- ( ( ( 1r ` R ) e. B /\ ( 0g ` R ) e. B /\ ( 1r ` R ) =/= ( 0g ` R ) ) -> E. x e. B E. y e. B -. x = y ) |
15 |
6 8 9 14
|
syl3anc |
|- ( ( R e. Ring /\ ( 1r ` R ) =/= ( 0g ` R ) ) -> E. x e. B E. y e. B -. x = y ) |
16 |
15
|
ex |
|- ( R e. Ring -> ( ( 1r ` R ) =/= ( 0g ` R ) -> E. x e. B E. y e. B -. x = y ) ) |
17 |
1 2 3
|
ring1eq0 |
|- ( ( R e. Ring /\ x e. B /\ y e. B ) -> ( ( 1r ` R ) = ( 0g ` R ) -> x = y ) ) |
18 |
17
|
3expb |
|- ( ( R e. Ring /\ ( x e. B /\ y e. B ) ) -> ( ( 1r ` R ) = ( 0g ` R ) -> x = y ) ) |
19 |
18
|
necon3bd |
|- ( ( R e. Ring /\ ( x e. B /\ y e. B ) ) -> ( -. x = y -> ( 1r ` R ) =/= ( 0g ` R ) ) ) |
20 |
19
|
rexlimdvva |
|- ( R e. Ring -> ( E. x e. B E. y e. B -. x = y -> ( 1r ` R ) =/= ( 0g ` R ) ) ) |
21 |
16 20
|
impbid |
|- ( R e. Ring -> ( ( 1r ` R ) =/= ( 0g ` R ) <-> E. x e. B E. y e. B -. x = y ) ) |
22 |
1
|
fvexi |
|- B e. _V |
23 |
|
1sdom |
|- ( B e. _V -> ( 1o ~< B <-> E. x e. B E. y e. B -. x = y ) ) |
24 |
22 23
|
ax-mp |
|- ( 1o ~< B <-> E. x e. B E. y e. B -. x = y ) |
25 |
21 24
|
bitr4di |
|- ( R e. Ring -> ( ( 1r ` R ) =/= ( 0g ` R ) <-> 1o ~< B ) ) |
26 |
|
1onn |
|- 1o e. _om |
27 |
|
sucdom |
|- ( 1o e. _om -> ( 1o ~< B <-> suc 1o ~<_ B ) ) |
28 |
26 27
|
ax-mp |
|- ( 1o ~< B <-> suc 1o ~<_ B ) |
29 |
|
df-2o |
|- 2o = suc 1o |
30 |
29
|
breq1i |
|- ( 2o ~<_ B <-> suc 1o ~<_ B ) |
31 |
28 30
|
bitr4i |
|- ( 1o ~< B <-> 2o ~<_ B ) |
32 |
25 31
|
bitrdi |
|- ( R e. Ring -> ( ( 1r ` R ) =/= ( 0g ` R ) <-> 2o ~<_ B ) ) |
33 |
32
|
pm5.32i |
|- ( ( R e. Ring /\ ( 1r ` R ) =/= ( 0g ` R ) ) <-> ( R e. Ring /\ 2o ~<_ B ) ) |
34 |
4 33
|
bitri |
|- ( R e. NzRing <-> ( R e. Ring /\ 2o ~<_ B ) ) |