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Theorem aptisr 6665
Description: Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.)
Assertion
Ref Expression
aptisr ((A R B 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.
StepHypRef Expression
1 df-nr 6615 . . 3 R = ((P × P) / ~R )
2 breq1 3758 . . . . . 6 ([⟨x, y⟩] ~R = A → ([⟨x, y⟩] ~R <R [⟨z, w⟩] ~RA <R [⟨z, w⟩] ~R ))
3 breq2 3759 . . . . . 6 ([⟨x, y⟩] ~R = A → ([⟨z, w⟩] ~R <R [⟨x, y⟩] ~R ↔ [⟨z, w⟩] ~R <R A))
42, 3orbi12d 706 . . . . 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)))
54notbid 591 . . . 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 2043 . . . 4 ([⟨x, y⟩] ~R = A → ([⟨x, y⟩] ~R = [⟨z, w⟩] ~RA = [⟨z, w⟩] ~R ))
75, 6imbi12d 223 . . 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 3759 . . . . . 6 ([⟨z, w⟩] ~R = B → (A <R [⟨z, w⟩] ~RA <R B))
9 breq1 3758 . . . . . 6 ([⟨z, w⟩] ~R = B → ([⟨z, w⟩] ~R <R AB <R A))
108, 9orbi12d 706 . . . . 5 ([⟨z, w⟩] ~R = B → ((A <R [⟨z, w⟩] ~R [⟨z, w⟩] ~R <R A) ↔ (A <R B B <R A)))
1110notbid 591 . . . 4 ([⟨z, w⟩] ~R = B → (¬ (A <R [⟨z, w⟩] ~R [⟨z, w⟩] ~R <R A) ↔ ¬ (A <R B B <R A)))
12 eqeq2 2046 . . . 4 ([⟨z, w⟩] ~R = B → (A = [⟨z, w⟩] ~RA = B))
1311, 12imbi12d 223 . . 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 6552 . . . . . . . . 9 ((y P z P) → (y +P z) = (z +P y))
1514ad2ant2lr 479 . . . . . . . 8 (((x P y P) (z P w P)) → (y +P z) = (z +P y))
16 addcomprg 6552 . . . . . . . . 9 ((x P w P) → (x +P w) = (w +P x))
1716ad2ant2rl 480 . . . . . . . 8 (((x P y P) (z P w P)) → (x +P w) = (w +P x))
1815, 17breq12d 3768 . . . . . . 7 (((x P y P) (z P w P)) → ((y +P z)<P (x +P w) ↔ (z +P y)<P (w +P x)))
1918orbi2d 703 . . . . . 6 (((x P y P) (z P w 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))))
2019notbid 591 . . . . 5 (((x P y P) (z P w 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 6520 . . . . . . 7 ((x P w P) → (x +P w) P)
2221ad2ant2rl 480 . . . . . 6 (((x P y P) (z P w P)) → (x +P w) P)
23 addclpr 6520 . . . . . . 7 ((y P z P) → (y +P z) P)
2423ad2ant2lr 479 . . . . . 6 (((x P y P) (z P w P)) → (y +P z) P)
25 aptipr 6611 . . . . . . 7 (((x +P w) P (y +P z) P ¬ ((x +P w)<P (y +P z) (y +P z)<P (x +P w))) → (x +P w) = (y +P z))
26253expia 1105 . . . . . 6 (((x +P w) P (y +P z) P) → (¬ ((x +P w)<P (y +P z) (y +P z)<P (x +P w)) → (x +P w) = (y +P z)))
2722, 24, 26syl2anc 391 . . . . 5 (((x P y P) (z P w P)) → (¬ ((x +P w)<P (y +P z) (y +P z)<P (x +P w)) → (x +P w) = (y +P z)))
2820, 27sylbird 159 . . . 4 (((x P y P) (z P w P)) → (¬ ((x +P w)<P (y +P z) (z +P y)<P (w +P x)) → (x +P w) = (y +P z)))
29 ltsrprg 6635 . . . . . 6 (((x P y P) (z P w P)) → ([⟨x, y⟩] ~R <R [⟨z, w⟩] ~R ↔ (x +P w)<P (y +P z)))
30 ltsrprg 6635 . . . . . . 7 (((z P w P) (x P y P)) → ([⟨z, w⟩] ~R <R [⟨x, y⟩] ~R ↔ (z +P y)<P (w +P x)))
3130ancoms 255 . . . . . 6 (((x P y P) (z P w P)) → ([⟨z, w⟩] ~R <R [⟨x, y⟩] ~R ↔ (z +P y)<P (w +P x)))
3229, 31orbi12d 706 . . . . 5 (((x P y P) (z P w 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))))
3332notbid 591 . . . 4 (((x P y P) (z P w 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 6624 . . . 4 (((x P y P) (z P w P)) → ([⟨x, y⟩] ~R = [⟨z, w⟩] ~R ↔ (x +P w) = (y +P z)))
3528, 33, 343imtr4d 192 . . 3 (((x P y P) (z P w P)) → (¬ ([⟨x, y⟩] ~R <R [⟨z, w⟩] ~R [⟨z, w⟩] ~R <R [⟨x, y⟩] ~R ) → [⟨x, y⟩] ~R = [⟨z, w⟩] ~R ))
361, 7, 13, 352ecoptocl 6130 . 2 ((A R B R) → (¬ (A <R B B <R A) → A = B))
37363impia 1100 1 ((A R B R ¬ (A <R B B <R A)) → A = B)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628   w3a 884   = wceq 1242   wcel 1390  cop 3370   class class class wbr 3755  (class class class)co 5455  [cec 6040  Pcnp 6275   +P cpp 6277  <P cltp 6279   ~R cer 6280  Rcnr 6281   <R cltr 6287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-iplp 6450  df-iltp 6452  df-enr 6614  df-nr 6615  df-ltr 6618
This theorem is referenced by:  axpre-apti  6729
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