Metamath Proof Explorer

Theorem noseponlem

Description: Lemma for nosepon . Consider a case of proper subset domain. (Contributed by Scott Fenton, 21-Sep-2020)

Ref Expression
Assertion noseponlem 
|- ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> -. A. x e. On ( A ` x ) = ( B ` x ) )

Proof

Step Hyp Ref Expression
1 nodmon 
 |-  ( A e. No -> dom A e. On )
2 1 3ad2ant1 
 |-  ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> dom A e. On )
3 nodmord 
 |-  ( A e. No -> Ord dom A )
4 ordirr 
 |-  ( Ord dom A -> -. dom A e. dom A )
5 3 4 syl 
 |-  ( A e. No -> -. dom A e. dom A )
6 5 3ad2ant1 
 |-  ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> -. dom A e. dom A )
7 ndmfv 
 |-  ( -. dom A e. dom A -> ( A ` dom A ) = (/) )
8 6 7 syl 
 |-  ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> ( A ` dom A ) = (/) )
9 nosgnn0 
 |-  -. (/) e. { 1o , 2o }
10 elno3 
 |-  ( B e. No <-> ( B : dom B --> { 1o , 2o } /\ dom B e. On ) )
11 10 simplbi 
 |-  ( B e. No -> B : dom B --> { 1o , 2o } )
12 11 3ad2ant2 
 |-  ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> B : dom B --> { 1o , 2o } )
13 simp3 
 |-  ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> dom A e. dom B )
14 12 13 ffvelrnd 
 |-  ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> ( B ` dom A ) e. { 1o , 2o } )
15 eleq1 
 |-  ( ( B ` dom A ) = (/) -> ( ( B ` dom A ) e. { 1o , 2o } <-> (/) e. { 1o , 2o } ) )
16 14 15 syl5ibcom 
 |-  ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> ( ( B ` dom A ) = (/) -> (/) e. { 1o , 2o } ) )
17 9 16 mtoi 
 |-  ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> -. ( B ` dom A ) = (/) )
18 17 neqned 
 |-  ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> ( B ` dom A ) =/= (/) )
19 18 necomd 
 |-  ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> (/) =/= ( B ` dom A ) )
20 8 19 eqnetrd 
 |-  ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> ( A ` dom A ) =/= ( B ` dom A ) )
21 fveq2 
 |-  ( x = dom A -> ( A ` x ) = ( A ` dom A ) )
22 fveq2 
 |-  ( x = dom A -> ( B ` x ) = ( B ` dom A ) )
23 21 22 neeq12d 
 |-  ( x = dom A -> ( ( A ` x ) =/= ( B ` x ) <-> ( A ` dom A ) =/= ( B ` dom A ) ) )
24 23 rspcev 
 |-  ( ( dom A e. On /\ ( A ` dom A ) =/= ( B ` dom A ) ) -> E. x e. On ( A ` x ) =/= ( B ` x ) )
25 2 20 24 syl2anc 
 |-  ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> E. x e. On ( A ` x ) =/= ( B ` x ) )
26 df-ne 
 |-  ( ( A ` x ) =/= ( B ` x ) <-> -. ( A ` x ) = ( B ` x ) )
27 26 rexbii 
 |-  ( E. x e. On ( A ` x ) =/= ( B ` x ) <-> E. x e. On -. ( A ` x ) = ( B ` x ) )
28 rexnal 
 |-  ( E. x e. On -. ( A ` x ) = ( B ` x ) <-> -. A. x e. On ( A ` x ) = ( B ` x ) )
29 27 28 bitri 
 |-  ( E. x e. On ( A ` x ) =/= ( B ` x ) <-> -. A. x e. On ( A ` x ) = ( B ` x ) )
30 25 29 sylib 
 |-  ( ( A e. No /\ B e. No /\ dom A e. dom B ) -> -. A. x e. On ( A ` x ) = ( B ` x ) )