Theorem ltntri 8154

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 8147	. . 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 8147	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> B <_ A )
7	1, 2	letri3d 8142	. . 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 2167   class class class wbr 4033   RRcr 7878   < clt 8061   <_ cle 8062

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-pre-ltirr 7991  ax-pre-apti 7994

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-xp 4669  df-cnv 4671  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067

This theorem is referenced by: (None)