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Metamath Proof Explorer

Theorem uztric

Description: Totality of the ordering relation on integers, stated in terms of upper integers. (Contributed by NM, 6-Jul-2005) (Revised by Mario Carneiro, 25-Jun-2013)

Ref Expression
Assertion uztric 
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( N e. ( ZZ>= ` M ) \/ M e. ( ZZ>= ` N ) ) )

Proof

Step Hyp Ref Expression
1 zre 
 |-  ( M e. ZZ -> M e. RR )
2 zre 
 |-  ( N e. ZZ -> N e. RR )
3 letric 
 |-  ( ( M e. RR /\ N e. RR ) -> ( M <_ N \/ N <_ M ) )
4 1 2 3 syl2an 
 |-  ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N \/ N <_ M ) )
5 eluz 
 |-  ( ( M e. ZZ /\ N e. ZZ ) -> ( N e. ( ZZ>= ` M ) <-> M <_ N ) )
6 eluz 
 |-  ( ( N e. ZZ /\ M e. ZZ ) -> ( M e. ( ZZ>= ` N ) <-> N <_ M ) )
7 6 ancoms 
 |-  ( ( M e. ZZ /\ N e. ZZ ) -> ( M e. ( ZZ>= ` N ) <-> N <_ M ) )
8 5 7 orbi12d 
 |-  ( ( M e. ZZ /\ N e. ZZ ) -> ( ( N e. ( ZZ>= ` M ) \/ M e. ( ZZ>= ` N ) ) <-> ( M <_ N \/ N <_ M ) ) )
9 4 8 mpbird 
 |-  ( ( M e. ZZ /\ N e. ZZ ) -> ( N e. ( ZZ>= ` M ) \/ M e. ( ZZ>= ` N ) ) )