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Theorem aptisr 7551
Description: Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.)
Assertion
Ref Expression
aptisr ((𝐴R𝐵R ∧ ¬ (𝐴 <R 𝐵𝐵 <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 7499 . . 3 R = ((P × P) / ~R )
2 breq1 3900 . . . . . 6 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R𝐴 <R [⟨𝑧, 𝑤⟩] ~R ))
3 breq2 3901 . . . . . 6 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ [⟨𝑧, 𝑤⟩] ~R <R 𝐴))
42, 3orbi12d 765 . . . . 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 639 . . . 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 2122 . . . 4 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R𝐴 = [⟨𝑧, 𝑤⟩] ~R ))
75, 6imbi12d 233 . . 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 3901 . . . . . 6 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 <R [⟨𝑧, 𝑤⟩] ~R𝐴 <R 𝐵))
9 breq1 3900 . . . . . 6 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ([⟨𝑧, 𝑤⟩] ~R <R 𝐴𝐵 <R 𝐴))
108, 9orbi12d 765 . . . . 5 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((𝐴 <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝐴) ↔ (𝐴 <R 𝐵𝐵 <R 𝐴)))
1110notbid 639 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (¬ (𝐴 <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝐴) ↔ ¬ (𝐴 <R 𝐵𝐵 <R 𝐴)))
12 eqeq2 2125 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 = [⟨𝑧, 𝑤⟩] ~R𝐴 = 𝐵))
1311, 12imbi12d 233 . . 3 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((¬ (𝐴 <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝐴) → 𝐴 = [⟨𝑧, 𝑤⟩] ~R ) ↔ (¬ (𝐴 <R 𝐵𝐵 <R 𝐴) → 𝐴 = 𝐵)))
14 addcomprg 7350 . . . . . . . . 9 ((𝑦P𝑧P) → (𝑦 +P 𝑧) = (𝑧 +P 𝑦))
1514ad2ant2lr 499 . . . . . . . 8 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (𝑦 +P 𝑧) = (𝑧 +P 𝑦))
16 addcomprg 7350 . . . . . . . . 9 ((𝑥P𝑤P) → (𝑥 +P 𝑤) = (𝑤 +P 𝑥))
1716ad2ant2rl 500 . . . . . . . 8 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (𝑥 +P 𝑤) = (𝑤 +P 𝑥))
1815, 17breq12d 3910 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑦 +P 𝑧)<P (𝑥 +P 𝑤) ↔ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥)))
1918orbi2d 762 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (((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 639 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (¬ ((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 7309 . . . . . . 7 ((𝑥P𝑤P) → (𝑥 +P 𝑤) ∈ P)
2221ad2ant2rl 500 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (𝑥 +P 𝑤) ∈ P)
23 addclpr 7309 . . . . . . 7 ((𝑦P𝑧P) → (𝑦 +P 𝑧) ∈ P)
2423ad2ant2lr 499 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (𝑦 +P 𝑧) ∈ P)
25 aptipr 7413 . . . . . . 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 1166 . . . . . 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 406 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (¬ ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ∨ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤)) → (𝑥 +P 𝑤) = (𝑦 +P 𝑧)))
2820, 27sylbird 169 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (¬ ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ∨ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥)) → (𝑥 +P 𝑤) = (𝑦 +P 𝑧)))
29 ltsrprg 7519 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧)))
30 ltsrprg 7519 . . . . . . 7 (((𝑧P𝑤P) ∧ (𝑥P𝑦P)) → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥)))
3130ancoms 266 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥)))
3229, 31orbi12d 765 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (([⟨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 639 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (¬ ([⟨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 7508 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤) = (𝑦 +P 𝑧)))
3528, 33, 343imtr4d 202 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (¬ ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ) → [⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ))
361, 7, 13, 352ecoptocl 6483 . 2 ((𝐴R𝐵R) → (¬ (𝐴 <R 𝐵𝐵 <R 𝐴) → 𝐴 = 𝐵))
37363impia 1161 1 ((𝐴R𝐵R ∧ ¬ (𝐴 <R 𝐵𝐵 <R 𝐴)) → 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 680  w3a 945   = wceq 1314  wcel 1463  cop 3498   class class class wbr 3897  (class class class)co 5740  [cec 6393  Pcnp 7063   +P cpp 7065  <P cltp 7067   ~R cer 7068  Rcnr 7069   <R cltr 7075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238  df-iplp 7240  df-iltp 7242  df-enr 7498  df-nr 7499  df-ltr 7502
This theorem is referenced by:  axpre-apti  7657
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