Metamath Proof Explorer

Theorem xreqle

Description: Equality implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Assertion xreqle 
|- ( ( A e. RR* /\ A = B ) -> A <_ B )

Proof

Step Hyp Ref Expression
1 xrleid 
 |-  ( A e. RR* -> A <_ A )
2 1 adantr 
 |-  ( ( A e. RR* /\ A = B ) -> A <_ A )
3 simpr 
 |-  ( ( A e. RR* /\ A = B ) -> A = B )
4 breq2 
 |-  ( A = B -> ( A <_ A <-> A <_ B ) )
5 4 biimpac 
 |-  ( ( A <_ A /\ A = B ) -> A <_ B )
6 2 3 5 syl2anc 
 |-  ( ( A e. RR* /\ A = B ) -> A <_ B )