Theorem ltntri 8079

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 1005	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> -. B < A )
4	1, 2, 3	nltled 8072	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> A <_ B )
5		simpr1 1003	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> -. A < B )
6	2, 1, 5	nltled 8072	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> B <_ A )
7	1, 2	letri3d 8067	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )
8	4, 6, 7	mpbir2and 944	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> A = B )
9		simpr2 1004	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> -. A = B )
10	8, 9	pm2.65da 661	1 |- ( ( A e. RR /\ B e. RR ) -> -. ( -. A < B /\ -. A = B /\ -. B < A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   class class class wbr 4001   RRcr 7805   < clt 7986   <_ cle 7987

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4119  ax-pow 4172  ax-pr 4207  ax-un 4431  ax-setind 4534  ax-cnex 7897  ax-resscn 7898  ax-pre-ltirr 7918  ax-pre-apti 7921

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-br 4002  df-opab 4063  df-xp 4630  df-cnv 4632  df-pnf 7988  df-mnf 7989  df-xr 7990  df-ltxr 7991  df-le 7992

This theorem is referenced by: (None)