Theorem ltntri 7914

Description: Negative trichotomy property for real numbers. It is well known that we cannot prove real number trichotomy, A < B \/ A = B \/ B < A. Does that mean there is a pair of real numbers where none of those hold (that is, where we can refute each of those three relationships)? Actually, no, as shown here. This is another example of distinguishing between being unable to prove something, or being able to refute it. (Contributed by Jim Kingdon, 13-Aug-2023.)

Assertion

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

Proof of Theorem ltntri

Step	Hyp	Ref	Expression
1		simpll 519	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> A e. RR )
2		simplr 520	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> B e. RR )
3		simpr3 990	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> -. B < A )
4	1, 2, 3	nltled 7907	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> A <_ B )
5		simpr1 988	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> -. A < B )
6	2, 1, 5	nltled 7907	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> B <_ A )
7	1, 2	letri3d 7903	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )
8	4, 6, 7	mpbir2and 929	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> A = B )
9		simpr2 989	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> -. A = B )
10	8, 9	pm2.65da 651	1 |- ( ( A e. RR /\ B e. RR ) -> -. ( -. A < B /\ -. A = B /\ -. B < A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   /\ w3a 963   = wceq 1332   e. wcel 1481   class class class wbr 3937   RRcr 7643   < clt 7824   <_ cle 7825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-pre-ltirr 7756  ax-pre-apti 7759

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-xp 4553  df-cnv 4555  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830

This theorem is referenced by: (None)