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Theorem axpre-lttri 11203
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 11330. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 11227. (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 11169 . 2 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
2 elreal 11169 . 2 (𝐵 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐵)
3 breq1 5151 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝐴 <𝑦, 0R⟩))
4 eqeq1 2739 . . . . 5 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ 𝐴 = ⟨𝑦, 0R⟩))
5 breq2 5152 . . . . 5 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ <𝑥, 0R⟩ ↔ ⟨𝑦, 0R⟩ < 𝐴))
64, 5orbi12d 918 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ (𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴)))
76notbid 318 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → (¬ (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ ¬ (𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴)))
83, 7bibi12d 345 . 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 5152 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <𝑦, 0R⟩ ↔ 𝐴 < 𝐵))
10 eqeq2 2747 . . . . 5 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 = ⟨𝑦, 0R⟩ ↔ 𝐴 = 𝐵))
11 breq1 5151 . . . . 5 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ < 𝐴𝐵 < 𝐴))
1210, 11orbi12d 918 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴) ↔ (𝐴 = 𝐵𝐵 < 𝐴)))
1312notbid 318 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → (¬ (𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴) ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))
149, 13bibi12d 345 . 2 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <𝑦, 0R⟩ ↔ ¬ (𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ < 𝐴)) ↔ (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴))))
15 ltsosr 11132 . . . 4 <R Or R
16 sotric 5626 . . . 4 (( <R Or R ∧ (𝑥R𝑦R)) → (𝑥 <R 𝑦 ↔ ¬ (𝑥 = 𝑦𝑦 <R 𝑥)))
1715, 16mpan 690 . . 3 ((𝑥R𝑦R) → (𝑥 <R 𝑦 ↔ ¬ (𝑥 = 𝑦𝑦 <R 𝑥)))
18 ltresr 11178 . . 3 (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
19 vex 3482 . . . . . 6 𝑥 ∈ V
2019eqresr 11175 . . . . 5 (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ 𝑥 = 𝑦)
21 ltresr 11178 . . . . 5 (⟨𝑦, 0R⟩ <𝑥, 0R⟩ ↔ 𝑦 <R 𝑥)
2220, 21orbi12i 914 . . . 4 ((⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ (𝑥 = 𝑦𝑦 <R 𝑥))
2322notbii 320 . . 3 (¬ (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩) ↔ ¬ (𝑥 = 𝑦𝑦 <R 𝑥))
2417, 18, 233bitr4g 314 . 2 ((𝑥R𝑦R) → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ ¬ (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <𝑥, 0R⟩)))
251, 2, 8, 14, 242gencl 3522 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵𝐵 < 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wcel 2106  cop 4637   class class class wbr 5148   Or wor 5596  Rcnr 10903  0Rc0r 10904   <R cltr 10909  cr 11152   < cltrr 11157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510  df-er 8744  df-ec 8746  df-qs 8750  df-ni 10910  df-pli 10911  df-mi 10912  df-lti 10913  df-plpq 10946  df-mpq 10947  df-ltpq 10948  df-enq 10949  df-nq 10950  df-erq 10951  df-plq 10952  df-mq 10953  df-1nq 10954  df-rq 10955  df-ltnq 10956  df-np 11019  df-1p 11020  df-plp 11021  df-ltp 11023  df-enr 11093  df-nr 11094  df-ltr 11097  df-0r 11098  df-r 11163  df-lt 11166
This theorem is referenced by: (None)
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