Theorem aptisr 7894

Description: Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.)

Assertion

Ref	Expression
aptisr	|- ( ( A e. R. /\ B e. R. /\ -. ( A <R B \/ B <R A ) ) -> A = B )

Proof of Theorem aptisr

Dummy variables w x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nr 7842	. . 3 |- R. = ( ( P. X. P. ) /. ~R )
2		breq1 4048	. . . . . 6 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> A <R [ <. z , w >. ] ~R ) )
3		breq2 4049	. . . . . 6 |- ( [ <. x , y >. ] ~R = A -> ( [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R <-> [ <. z , w >. ] ~R <R A ) )
4	2, 3	orbi12d 795	. . . . 5 |- ( [ <. x , y >. ] ~R = A -> ( ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R ) <-> ( A <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R A ) ) )
5	4	notbid 669	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( -. ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R ) <-> -. ( A <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R A ) ) )
6		eqeq1 2212	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R = [ <. z , w >. ] ~R <-> A = [ <. z , w >. ] ~R ) )
7	5, 6	imbi12d 234	. . 3 |- ( [ <. x , y >. ] ~R = A -> ( ( -. ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R ) -> [ <. x , y >. ] ~R = [ <. z , w >. ] ~R ) <-> ( -. ( A <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R A ) -> A = [ <. z , w >. ] ~R ) ) )
8		breq2 4049	. . . . . 6 |- ( [ <. z , w >. ] ~R = B -> ( A <R [ <. z , w >. ] ~R <-> A <R B ) )
9		breq1 4048	. . . . . 6 |- ( [ <. z , w >. ] ~R = B -> ( [ <. z , w >. ] ~R <R A <-> B <R A ) )
10	8, 9	orbi12d 795	. . . . 5 |- ( [ <. z , w >. ] ~R = B -> ( ( A <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R A ) <-> ( A <R B \/ B <R A ) ) )
11	10	notbid 669	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( -. ( A <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R A ) <-> -. ( A <R B \/ B <R A ) ) )
12		eqeq2 2215	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( A = [ <. z , w >. ] ~R <-> A = B ) )
13	11, 12	imbi12d 234	. . 3 |- ( [ <. z , w >. ] ~R = B -> ( ( -. ( A <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R A ) -> A = [ <. z , w >. ] ~R ) <-> ( -. ( A <R B \/ B <R A ) -> A = B ) ) )
14		addcomprg 7693	. . . . . . . . 9 |- ( ( y e. P. /\ z e. P. ) -> ( y +P. z ) = ( z +P. y ) )
15	14	ad2ant2lr 510	. . . . . . . 8 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y +P. z ) = ( z +P. y ) )
16		addcomprg 7693	. . . . . . . . 9 |- ( ( x e. P. /\ w e. P. ) -> ( x +P. w ) = ( w +P. x ) )
17	16	ad2ant2rl 511	. . . . . . . 8 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x +P. w ) = ( w +P. x ) )
18	15, 17	breq12d 4058	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( y +P. z ) <P ( x +P. w ) <-> ( z +P. y ) <P ( w +P. x ) ) )
19	18	orbi2d 792	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( ( x +P. w ) <P ( y +P. z ) \/ ( y +P. z ) <P ( x +P. w ) ) <-> ( ( x +P. w ) <P ( y +P. z ) \/ ( z +P. y ) <P ( w +P. x ) ) ) )
20	19	notbid 669	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( -. ( ( x +P. w ) <P ( y +P. z ) \/ ( y +P. z ) <P ( x +P. w ) ) <-> -. ( ( x +P. w ) <P ( y +P. z ) \/ ( z +P. y ) <P ( w +P. x ) ) ) )
21		addclpr 7652	. . . . . . 7 |- ( ( x e. P. /\ w e. P. ) -> ( x +P. w ) e. P. )
22	21	ad2ant2rl 511	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x +P. w ) e. P. )
23		addclpr 7652	. . . . . . 7 |- ( ( y e. P. /\ z e. P. ) -> ( y +P. z ) e. P. )
24	23	ad2ant2lr 510	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y +P. z ) e. P. )
25		aptipr 7756	. . . . . . 7 |- ( ( ( x +P. w ) e. P. /\ ( y +P. z ) e. P. /\ -. ( ( x +P. w ) <P ( y +P. z ) \/ ( y +P. z ) <P ( x +P. w ) ) ) -> ( x +P. w ) = ( y +P. z ) )
26	25	3expia 1208	. . . . . 6 |- ( ( ( x +P. w ) e. P. /\ ( y +P. z ) e. P. ) -> ( -. ( ( x +P. w ) <P ( y +P. z ) \/ ( y +P. z ) <P ( x +P. w ) ) -> ( x +P. w ) = ( y +P. z ) ) )
27	22, 24, 26	syl2anc 411	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( -. ( ( x +P. w ) <P ( y +P. z ) \/ ( y +P. z ) <P ( x +P. w ) ) -> ( x +P. w ) = ( y +P. z ) ) )
28	20, 27	sylbird 170	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( -. ( ( x +P. w ) <P ( y +P. z ) \/ ( z +P. y ) <P ( w +P. x ) ) -> ( x +P. w ) = ( y +P. z ) ) )
29		ltsrprg 7862	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> ( x +P. w ) <P ( y +P. z ) ) )
30		ltsrprg 7862	. . . . . . 7 |- ( ( ( z e. P. /\ w e. P. ) /\ ( x e. P. /\ y e. P. ) ) -> ( [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R <-> ( z +P. y ) <P ( w +P. x ) ) )
31	30	ancoms 268	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R <-> ( z +P. y ) <P ( w +P. x ) ) )
32	29, 31	orbi12d 795	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R ) <-> ( ( x +P. w ) <P ( y +P. z ) \/ ( z +P. y ) <P ( w +P. x ) ) ) )
33	32	notbid 669	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( -. ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R ) <-> -. ( ( x +P. w ) <P ( y +P. z ) \/ ( z +P. y ) <P ( w +P. x ) ) ) )
34		enreceq 7851	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( [ <. x , y >. ] ~R = [ <. z , w >. ] ~R <-> ( x +P. w ) = ( y +P. z ) ) )
35	28, 33, 34	3imtr4d 203	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( -. ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R ) -> [ <. x , y >. ] ~R = [ <. z , w >. ] ~R ) )
36	1, 7, 13, 35	2ecoptocl 6712	. 2 |- ( ( A e. R. /\ B e. R. ) -> ( -. ( A <R B \/ B <R A ) -> A = B ) )
37	36	3impia 1203	1 |- ( ( A e. R. /\ B e. R. /\ -. ( A <R B \/ B <R A ) ) -> A = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   /\ w3a 981   = wceq 1373   e. wcel 2176   <.cop 3636   class class class wbr 4045  (class class class)co 5946   [cec 6620   P.cnp 7406   +P. cpp 7408   <P cltp 7410   ~R cer 7411   R.cnr 7412   <R cltr 7418

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-eprel 4337  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-irdg 6458  df-1o 6504  df-2o 6505  df-oadd 6508  df-omul 6509  df-er 6622  df-ec 6624  df-qs 6628  df-ni 7419  df-pli 7420  df-mi 7421  df-lti 7422  df-plpq 7459  df-mpq 7460  df-enq 7462  df-nqqs 7463  df-plqqs 7464  df-mqqs 7465  df-1nqqs 7466  df-rq 7467  df-ltnqqs 7468  df-enq0 7539  df-nq0 7540  df-0nq0 7541  df-plq0 7542  df-mq0 7543  df-inp 7581  df-iplp 7583  df-iltp 7585  df-enr 7841  df-nr 7842  df-ltr 7845

This theorem is referenced by:  axpre-apti  8000