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Theorem axpre-lttri 10575
Description: Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axlttri 10700. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 10599. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
axpre-lttri ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))

Proof of Theorem axpre-lttri
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elreal 10541 . 2 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
2 elreal 10541 . 2 (𝐵 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐵)
3 breq1 5060 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝐴 <𝑦, 0R⟩))
4 eqeq1 2822 . . . . 5 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ 𝐴 = ⟨𝑦, 0R⟩))
5 breq2 5061 . . . . 5 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ <𝑥, 0R⟩ ↔ ⟨𝑦, 0R⟩ < 𝐴))
64, 5orbi12d 912 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ (𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴)))
76notbid 319 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → (¬ (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ ¬ (𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴)))
83, 7bibi12d 347 . 2 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ ¬ (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩)) ↔ (𝐴 <𝑦, 0R⟩ ↔ ¬ (𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴))))
9 breq2 5061 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <𝑦, 0R⟩ ↔ 𝐴 < 𝐵))
10 eqeq2 2830 . . . . 5 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 = ⟨𝑦, 0R⟩ ↔ 𝐴 = 𝐵))
11 breq1 5060 . . . . 5 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ < 𝐴𝐵 < 𝐴))
1210, 11orbi12d 912 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴) ↔ (𝐴 = 𝐵𝐵 < 𝐴)))
1312notbid 319 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → (¬ (𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴) ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))
149, 13bibi12d 347 . 2 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <𝑦, 0R⟩ ↔ ¬ (𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴)) ↔ (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴))))
15 ltsosr 10504 . . . 4 <R Or R
16 sotric 5494 . . . 4 (( <R Or R ∧ (𝑥R𝑦R)) → (𝑥 <R 𝑦 ↔ ¬ (𝑥 = 𝑦𝑦 <R 𝑥)))
1715, 16mpan 686 . . 3 ((𝑥R𝑦R) → (𝑥 <R 𝑦 ↔ ¬ (𝑥 = 𝑦𝑦 <R 𝑥)))
18 ltresr 10550 . . 3 (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
19 vex 3495 . . . . . 6 𝑥 ∈ V
2019eqresr 10547 . . . . 5 (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ 𝑥 = 𝑦)
21 ltresr 10550 . . . . 5 (⟨𝑦, 0R⟩ <𝑥, 0R⟩ ↔ 𝑦 <R 𝑥)
2220, 21orbi12i 908 . . . 4 ((⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ (𝑥 = 𝑦𝑦 <R 𝑥))
2322notbii 321 . . 3 (¬ (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ ¬ (𝑥 = 𝑦𝑦 <R 𝑥))
2417, 18, 233bitr4g 315 . 2 ((𝑥R𝑦R) → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ ¬ (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩)))
251, 2, 8, 14, 242gencl 3533 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  cop 4563   class class class wbr 5057   Or wor 5466  Rcnr 10275  0Rc0r 10276   <R cltr 10281  cr 10524   < cltrr 10529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-omul 8096  df-er 8278  df-ec 8280  df-qs 8284  df-ni 10282  df-pli 10283  df-mi 10284  df-lti 10285  df-plpq 10318  df-mpq 10319  df-ltpq 10320  df-enq 10321  df-nq 10322  df-erq 10323  df-plq 10324  df-mq 10325  df-1nq 10326  df-rq 10327  df-ltnq 10328  df-np 10391  df-1p 10392  df-plp 10393  df-ltp 10395  df-enr 10465  df-nr 10466  df-ltr 10469  df-0r 10470  df-r 10535  df-lt 10538
This theorem is referenced by: (None)
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