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Metamath Proof Explorer

Theorem axlttri

Description: Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-lttri with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005)

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

Proof

Step Hyp Ref Expression
1 ax-pre-lttri 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A  -. ( A = B \/ B
2 ltxrlt 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A
3 ltxrlt 
 |-  ( ( B e. RR /\ A e. RR ) -> ( B < A <-> B
4 3 ancoms 
 |-  ( ( A e. RR /\ B e. RR ) -> ( B < A <-> B
5 4 orbi2d 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( A = B \/ B < A ) <-> ( A = B \/ B
6 5 notbid 
 |-  ( ( A e. RR /\ B e. RR ) -> ( -. ( A = B \/ B < A ) <-> -. ( A = B \/ B
7 1 2 6 3bitr4d 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -. ( A = B \/ B < A ) ) )