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Theorem axpre-apti 7016
Description: Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-apti 7056.

(Contributed by Jim Kingdon, 29-Jan-2020.) (New usage is discouraged.)

Assertion
Ref Expression
axpre-apti ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 < 𝐵𝐵 < 𝐴)) → 𝐴 = 𝐵)

Proof of Theorem axpre-apti
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elreal 6962 . . 3 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
2 elreal 6962 . . 3 (𝐵 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐵)
3 breq1 3794 . . . . . 6 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝐴 <𝑦, 0R⟩))
4 breq2 3795 . . . . . 6 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ <𝑥, 0R⟩ ↔ ⟨𝑦, 0R⟩ < 𝐴))
53, 4orbi12d 717 . . . . 5 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ (𝐴 <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴)))
65notbid 602 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (¬ (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ ¬ (𝐴 <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴)))
7 eqeq1 2062 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ 𝐴 = ⟨𝑦, 0R⟩))
86, 7imbi12d 227 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → ((¬ (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) → ⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩) ↔ (¬ (𝐴 <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴) → 𝐴 = ⟨𝑦, 0R⟩)))
9 breq2 3795 . . . . . 6 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <𝑦, 0R⟩ ↔ 𝐴 < 𝐵))
10 breq1 3794 . . . . . 6 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ < 𝐴𝐵 < 𝐴))
119, 10orbi12d 717 . . . . 5 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴) ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
1211notbid 602 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (¬ (𝐴 <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴) ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
13 eqeq2 2065 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 = ⟨𝑦, 0R⟩ ↔ 𝐴 = 𝐵))
1412, 13imbi12d 227 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → ((¬ (𝐴 <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴) → 𝐴 = ⟨𝑦, 0R⟩) ↔ (¬ (𝐴 < 𝐵𝐵 < 𝐴) → 𝐴 = 𝐵)))
15 aptisr 6920 . . . . 5 ((𝑥R𝑦R ∧ ¬ (𝑥 <R 𝑦𝑦 <R 𝑥)) → 𝑥 = 𝑦)
16153expia 1117 . . . 4 ((𝑥R𝑦R) → (¬ (𝑥 <R 𝑦𝑦 <R 𝑥) → 𝑥 = 𝑦))
17 ltresr 6972 . . . . . 6 (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
18 ltresr 6972 . . . . . 6 (⟨𝑦, 0R⟩ <𝑥, 0R⟩ ↔ 𝑦 <R 𝑥)
1917, 18orbi12i 691 . . . . 5 ((⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ (𝑥 <R 𝑦𝑦 <R 𝑥))
2019notbii 604 . . . 4 (¬ (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ ¬ (𝑥 <R 𝑦𝑦 <R 𝑥))
21 vex 2577 . . . . 5 𝑥 ∈ V
2221eqresr 6969 . . . 4 (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ 𝑥 = 𝑦)
2316, 20, 223imtr4g 198 . . 3 ((𝑥R𝑦R) → (¬ (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) → ⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩))
241, 2, 8, 14, 232gencl 2604 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ (𝐴 < 𝐵𝐵 < 𝐴) → 𝐴 = 𝐵))
25243impia 1112 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 < 𝐵𝐵 < 𝐴)) → 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wo 639  w3a 896   = wceq 1259  wcel 1409  cop 3405   class class class wbr 3791  Rcnr 6452  0Rc0r 6453   <R cltr 6458  cr 6945   < cltrr 6950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-i1p 6622  df-iplp 6623  df-iltp 6625  df-enr 6868  df-nr 6869  df-ltr 6872  df-0r 6873  df-r 6956  df-lt 6959
This theorem is referenced by: (None)
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