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Theorem aptisr 6515
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 6465 . . 3 R = ((P × P) / ~R )
2 breq1 3730 . . . . . 6 ([⟨x, y⟩] ~R = A → ([⟨x, y⟩] ~R <R [⟨z, w⟩] ~RA <R [⟨z, w⟩] ~R ))
3 breq2 3731 . . . . . 6 ([⟨x, y⟩] ~R = A → ([⟨z, w⟩] ~R <R [⟨x, y⟩] ~R ↔ [⟨z, w⟩] ~R <R A))
42, 3orbi12d 691 . . . . 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 576 . . . 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 2019 . . . 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 3731 . . . . . 6 ([⟨z, w⟩] ~R = B → (A <R [⟨z, w⟩] ~RA <R B))
9 breq1 3730 . . . . . 6 ([⟨z, w⟩] ~R = B → ([⟨z, w⟩] ~R <R AB <R A))
108, 9orbi12d 691 . . . . 5 ([⟨z, w⟩] ~R = B → ((A <R [⟨z, w⟩] ~R [⟨z, w⟩] ~R <R A) ↔ (A <R B B <R A)))
1110notbid 576 . . . 4 ([⟨z, w⟩] ~R = B → (¬ (A <R [⟨z, w⟩] ~R [⟨z, w⟩] ~R <R A) ↔ ¬ (A <R B B <R A)))
12 eqeq2 2022 . . . 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 6403 . . . . . . . . 9 ((y P z P) → (y +P z) = (z +P y))
1514ad2ant2lr 464 . . . . . . . 8 (((x P y P) (z P w P)) → (y +P z) = (z +P y))
16 addcomprg 6403 . . . . . . . . 9 ((x P w P) → (x +P w) = (w +P x))
1716ad2ant2rl 465 . . . . . . . 8 (((x P y P) (z P w P)) → (x +P w) = (w +P x))
1815, 17breq12d 3740 . . . . . . 7 (((x P y P) (z P w P)) → ((y +P z)<P (x +P w) ↔ (z +P y)<P (w +P x)))
1918orbi2d 688 . . . . . 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 576 . . . . 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 6378 . . . . . . 7 ((x P w P) → (x +P w) P)
2221ad2ant2rl 465 . . . . . 6 (((x P y P) (z P w P)) → (x +P w) P)
23 addclpr 6378 . . . . . . 7 ((y P z P) → (y +P z) P)
2423ad2ant2lr 464 . . . . . 6 (((x P y P) (z P w P)) → (y +P z) P)
25 aptipr 6462 . . . . . . 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 1087 . . . . . 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 6485 . . . . . 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 6485 . . . . . . 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 691 . . . . 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 576 . . . 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 6474 . . . 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 6093 . 2 ((A R B R) → (¬ (A <R B B <R A) → A = B))
37363impia 1082 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 613   w3a 867   = wceq 1223   wcel 1366  cop 3342   class class class wbr 3727  (class class class)co 5424  [cec 6003  Pcnp 6137   +P cpp 6139  <P cltp 6141   ~R cer 6142  Rcnr 6143   <R cltr 6149
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 529  ax-in2 530  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-13 1377  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-coll 3835  ax-sep 3838  ax-nul 3846  ax-pow 3890  ax-pr 3907  ax-un 4108  ax-setind 4192  ax-iinf 4226
This theorem depends on definitions:  df-bi 110  df-dc 727  df-3or 868  df-3an 869  df-tru 1226  df-fal 1229  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ne 2179  df-ral 2280  df-rex 2281  df-reu 2282  df-rab 2284  df-v 2528  df-sbc 2733  df-csb 2821  df-dif 2888  df-un 2890  df-in 2892  df-ss 2899  df-nul 3193  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-uni 3544  df-int 3579  df-iun 3622  df-br 3728  df-opab 3782  df-mpt 3783  df-tr 3818  df-eprel 3989  df-id 3993  df-po 3996  df-iso 3997  df-iord 4041  df-on 4043  df-suc 4046  df-iom 4229  df-xp 4266  df-rel 4267  df-cnv 4268  df-co 4269  df-dm 4270  df-rn 4271  df-res 4272  df-ima 4273  df-iota 4782  df-fun 4819  df-fn 4820  df-f 4821  df-f1 4822  df-fo 4823  df-f1o 4824  df-fv 4825  df-ov 5427  df-oprab 5428  df-mpt2 5429  df-1st 5678  df-2nd 5679  df-recs 5830  df-irdg 5866  df-1o 5904  df-2o 5905  df-oadd 5908  df-omul 5909  df-er 6005  df-ec 6007  df-qs 6011  df-ni 6150  df-pli 6151  df-mi 6152  df-lti 6153  df-plpq 6189  df-mpq 6190  df-enq 6192  df-nqqs 6193  df-plqqs 6194  df-mqqs 6195  df-1nqqs 6196  df-rq 6197  df-ltnqqs 6198  df-enq0 6265  df-nq0 6266  df-0nq0 6267  df-plq0 6268  df-mq0 6269  df-inp 6306  df-iplp 6308  df-iltp 6310  df-enr 6464  df-nr 6465  df-ltr 6468
This theorem is referenced by:  axpre-apti  6574
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