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Theorem axpre-apti 7818
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 7860.

(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 7761 . . 3 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
2 elreal 7761 . . 3 (𝐵 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐵)
3 breq1 3980 . . . . . 6 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝐴 <𝑦, 0R⟩))
4 breq2 3981 . . . . . 6 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ <𝑥, 0R⟩ ↔ ⟨𝑦, 0R⟩ < 𝐴))
53, 4orbi12d 783 . . . . 5 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ (𝐴 <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴)))
65notbid 657 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (¬ (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ ¬ (𝐴 <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴)))
7 eqeq1 2171 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ 𝐴 = ⟨𝑦, 0R⟩))
86, 7imbi12d 233 . . 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 3981 . . . . . 6 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <𝑦, 0R⟩ ↔ 𝐴 < 𝐵))
10 breq1 3980 . . . . . 6 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ < 𝐴𝐵 < 𝐴))
119, 10orbi12d 783 . . . . 5 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴) ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
1211notbid 657 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (¬ (𝐴 <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴) ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
13 eqeq2 2174 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 = ⟨𝑦, 0R⟩ ↔ 𝐴 = 𝐵))
1412, 13imbi12d 233 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → ((¬ (𝐴 <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴) → 𝐴 = ⟨𝑦, 0R⟩) ↔ (¬ (𝐴 < 𝐵𝐵 < 𝐴) → 𝐴 = 𝐵)))
15 aptisr 7712 . . . . 5 ((𝑥R𝑦R ∧ ¬ (𝑥 <R 𝑦𝑦 <R 𝑥)) → 𝑥 = 𝑦)
16153expia 1194 . . . 4 ((𝑥R𝑦R) → (¬ (𝑥 <R 𝑦𝑦 <R 𝑥) → 𝑥 = 𝑦))
17 ltresr 7772 . . . . . 6 (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
18 ltresr 7772 . . . . . 6 (⟨𝑦, 0R⟩ <𝑥, 0R⟩ ↔ 𝑦 <R 𝑥)
1917, 18orbi12i 754 . . . . 5 ((⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ (𝑥 <R 𝑦𝑦 <R 𝑥))
2019notbii 658 . . . 4 (¬ (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ ¬ (𝑥 <R 𝑦𝑦 <R 𝑥))
21 vex 2725 . . . . 5 𝑥 ∈ V
2221eqresr 7769 . . . 4 (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ 𝑥 = 𝑦)
2316, 20, 223imtr4g 204 . . 3 ((𝑥R𝑦R) → (¬ (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) → ⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩))
241, 2, 8, 14, 232gencl 2755 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ (𝐴 < 𝐵𝐵 < 𝐴) → 𝐴 = 𝐵))
25243impia 1189 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 < 𝐵𝐵 < 𝐴)) → 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 698  w3a 967   = wceq 1342  wcel 2135  cop 3574   class class class wbr 3977  Rcnr 7230  0Rc0r 7231   <R cltr 7236  cr 7744   < cltrr 7749
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4092  ax-sep 4095  ax-nul 4103  ax-pow 4148  ax-pr 4182  ax-un 4406  ax-setind 4509  ax-iinf 4560
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2724  df-sbc 2948  df-csb 3042  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-nul 3406  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-int 3820  df-iun 3863  df-br 3978  df-opab 4039  df-mpt 4040  df-tr 4076  df-eprel 4262  df-id 4266  df-po 4269  df-iso 4270  df-iord 4339  df-on 4341  df-suc 4344  df-iom 4563  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-rn 4610  df-res 4611  df-ima 4612  df-iota 5148  df-fun 5185  df-fn 5186  df-f 5187  df-f1 5188  df-fo 5189  df-f1o 5190  df-fv 5191  df-ov 5840  df-oprab 5841  df-mpo 5842  df-1st 6101  df-2nd 6102  df-recs 6265  df-irdg 6330  df-1o 6376  df-2o 6377  df-oadd 6380  df-omul 6381  df-er 6493  df-ec 6495  df-qs 6499  df-ni 7237  df-pli 7238  df-mi 7239  df-lti 7240  df-plpq 7277  df-mpq 7278  df-enq 7280  df-nqqs 7281  df-plqqs 7282  df-mqqs 7283  df-1nqqs 7284  df-rq 7285  df-ltnqqs 7286  df-enq0 7357  df-nq0 7358  df-0nq0 7359  df-plq0 7360  df-mq0 7361  df-inp 7399  df-i1p 7400  df-iplp 7401  df-iltp 7403  df-enr 7659  df-nr 7660  df-ltr 7663  df-0r 7664  df-r 7755  df-lt 7758
This theorem is referenced by: (None)
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