Theorem ltntri 8133

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 527	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> A e. RR )
2		simplr 528	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> B e. RR )
3		simpr3 1007	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> -. B < A )
4	1, 2, 3	nltled 8126	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> A <_ B )
5		simpr1 1005	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> -. A < B )
6	2, 1, 5	nltled 8126	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> B <_ A )
7	1, 2	letri3d 8121	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )
8	4, 6, 7	mpbir2and 946	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> A = B )
9		simpr2 1006	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> -. A = B )
10	8, 9	pm2.65da 662	1 |- ( ( A e. RR /\ B e. RR ) -> -. ( -. A < B /\ -. A = B /\ -. B < A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2160   class class class wbr 4025   RRcr 7857   < clt 8040   <_ cle 8041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4143  ax-pow 4199  ax-pr 4234  ax-un 4458  ax-setind 4561  ax-cnex 7949  ax-resscn 7950  ax-pre-ltirr 7970  ax-pre-apti 7973

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2758  df-dif 3151  df-un 3153  df-in 3155  df-ss 3162  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-br 4026  df-opab 4087  df-xp 4657  df-cnv 4659  df-pnf 8042  df-mnf 8043  df-xr 8044  df-ltxr 8045  df-le 8046

This theorem is referenced by: (None)