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Metamath Proof Explorer

Theorem peano2uz

Description: Second Peano postulate for an upper set of integers. (Contributed by NM, 7-Sep-2005)

Ref Expression
Assertion peano2uz 
|- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )

Proof

Step Hyp Ref Expression
1 simp1 
 |-  ( ( M e. ZZ /\ N e. ZZ /\ M <_ N ) -> M e. ZZ )
2 peano2z 
 |-  ( N e. ZZ -> ( N + 1 ) e. ZZ )
3 2 3ad2ant2 
 |-  ( ( M e. ZZ /\ N e. ZZ /\ M <_ N ) -> ( N + 1 ) e. ZZ )
4 zre 
 |-  ( M e. ZZ -> M e. RR )
5 zre 
 |-  ( N e. ZZ -> N e. RR )
6 letrp1 
 |-  ( ( M e. RR /\ N e. RR /\ M <_ N ) -> M <_ ( N + 1 ) )
7 5 6 syl3an2 
 |-  ( ( M e. RR /\ N e. ZZ /\ M <_ N ) -> M <_ ( N + 1 ) )
8 4 7 syl3an1 
 |-  ( ( M e. ZZ /\ N e. ZZ /\ M <_ N ) -> M <_ ( N + 1 ) )
9 1 3 8 3jca 
 |-  ( ( M e. ZZ /\ N e. ZZ /\ M <_ N ) -> ( M e. ZZ /\ ( N + 1 ) e. ZZ /\ M <_ ( N + 1 ) ) )
10 eluz2 
 |-  ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
11 eluz2 
 |-  ( ( N + 1 ) e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ ( N + 1 ) e. ZZ /\ M <_ ( N + 1 ) ) )
12 9 10 11 3imtr4i 
 |-  ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )