Theorem ltntri 8454

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 531	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> A e. RR )
2		simplr 533	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> B e. RR )
3		simpr3 1036	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> -. B < A )
4	1, 2, 3	nltled 8447	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> A <_ B )
5		simpr1 1034	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> -. A < B )
6	2, 1, 5	nltled 8447	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> B <_ A )
7	1, 2	letri3d 8441	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )
8	4, 6, 7	mpbir2and 957	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> A = B )
9		simpr2 1035	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( -. A < B /\ -. A = B /\ -. B < A ) ) -> -. A = B )
10	8, 9	pm2.65da 671	1 |- ( ( A e. RR /\ B e. RR ) -> -. ( -. A < B /\ -. A = B /\ -. B < A ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4130   RRcr 8178   < clt 8360   <_ cle 8361

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-pre-ltirr 8291  ax-pre-apti 8294

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-cnv 4782  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366

This theorem is used by: (None)