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Metamath Proof Explorer

Theorem enqeq

Description: Corollary of nqereu : if two fractions are both reduced and equivalent, then they are equal. (Contributed by Mario Carneiro, 6-May-2013) (New usage is discouraged.)

Ref Expression
Assertion enqeq 
|- ( ( A e. Q. /\ B e. Q. /\ A ~Q B ) -> A = B )

Proof

Step Hyp Ref Expression
1 3simpa 
 |-  ( ( A e. Q. /\ B e. Q. /\ A ~Q B ) -> ( A e. Q. /\ B e. Q. ) )
2 elpqn 
 |-  ( B e. Q. -> B e. ( N. X. N. ) )
3 2 3ad2ant2 
 |-  ( ( A e. Q. /\ B e. Q. /\ A ~Q B ) -> B e. ( N. X. N. ) )
4 nqereu 
 |-  ( B e. ( N. X. N. ) -> E! x e. Q. x ~Q B )
5 reurmo 
 |-  ( E! x e. Q. x ~Q B -> E* x e. Q. x ~Q B )
6 3 4 5 3syl 
 |-  ( ( A e. Q. /\ B e. Q. /\ A ~Q B ) -> E* x e. Q. x ~Q B )
7 df-rmo 
 |-  ( E* x e. Q. x ~Q B <-> E* x ( x e. Q. /\ x ~Q B ) )
8 6 7 sylib 
 |-  ( ( A e. Q. /\ B e. Q. /\ A ~Q B ) -> E* x ( x e. Q. /\ x ~Q B ) )
9 3simpb 
 |-  ( ( A e. Q. /\ B e. Q. /\ A ~Q B ) -> ( A e. Q. /\ A ~Q B ) )
10 simp2 
 |-  ( ( A e. Q. /\ B e. Q. /\ A ~Q B ) -> B e. Q. )
11 enqer 
 |-  ~Q Er ( N. X. N. )
12 11 a1i 
 |-  ( ( A e. Q. /\ B e. Q. /\ A ~Q B ) -> ~Q Er ( N. X. N. ) )
13 12 3 erref 
 |-  ( ( A e. Q. /\ B e. Q. /\ A ~Q B ) -> B ~Q B )
14 10 13 jca 
 |-  ( ( A e. Q. /\ B e. Q. /\ A ~Q B ) -> ( B e. Q. /\ B ~Q B ) )
15 eleq1 
 |-  ( x = A -> ( x e. Q. <-> A e. Q. ) )
16 breq1 
 |-  ( x = A -> ( x ~Q B <-> A ~Q B ) )
17 15 16 anbi12d 
 |-  ( x = A -> ( ( x e. Q. /\ x ~Q B ) <-> ( A e. Q. /\ A ~Q B ) ) )
18 eleq1 
 |-  ( x = B -> ( x e. Q. <-> B e. Q. ) )
19 breq1 
 |-  ( x = B -> ( x ~Q B <-> B ~Q B ) )
20 18 19 anbi12d 
 |-  ( x = B -> ( ( x e. Q. /\ x ~Q B ) <-> ( B e. Q. /\ B ~Q B ) ) )
21 17 20 moi 
 |-  ( ( ( A e. Q. /\ B e. Q. ) /\ E* x ( x e. Q. /\ x ~Q B ) /\ ( ( A e. Q. /\ A ~Q B ) /\ ( B e. Q. /\ B ~Q B ) ) ) -> A = B )
22 1 8 9 14 21 syl112anc 
 |-  ( ( A e. Q. /\ B e. Q. /\ A ~Q B ) -> A = B )