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Theorem axpre-ltwlin 7803
Description: Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltwlin 7845. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.)
Assertion
Ref Expression
axpre-ltwlin ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐴 < 𝐶𝐶 < 𝐵)))

Proof of Theorem axpre-ltwlin
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elreal 7748 . 2 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
2 elreal 7748 . 2 (𝐵 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐵)
3 elreal 7748 . 2 (𝐶 ∈ ℝ ↔ ∃𝑧R𝑧, 0R⟩ = 𝐶)
4 breq1 3968 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝐴 <𝑦, 0R⟩))
5 breq1 3968 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑧, 0R⟩ ↔ 𝐴 <𝑧, 0R⟩))
65orbi1d 781 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ <𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <𝑦, 0R⟩) ↔ (𝐴 <𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <𝑦, 0R⟩)))
74, 6imbi12d 233 . 2 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ <𝑦, 0R⟩ → (⟨𝑥, 0R⟩ <𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <𝑦, 0R⟩)) ↔ (𝐴 <𝑦, 0R⟩ → (𝐴 <𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <𝑦, 0R⟩))))
8 breq2 3969 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <𝑦, 0R⟩ ↔ 𝐴 < 𝐵))
9 breq2 3969 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑧, 0R⟩ <𝑦, 0R⟩ ↔ ⟨𝑧, 0R⟩ < 𝐵))
109orbi2d 780 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <𝑦, 0R⟩) ↔ (𝐴 <𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ < 𝐵)))
118, 10imbi12d 233 . 2 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <𝑦, 0R⟩ → (𝐴 <𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <𝑦, 0R⟩)) ↔ (𝐴 < 𝐵 → (𝐴 <𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ < 𝐵))))
12 breq2 3969 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → (𝐴 <𝑧, 0R⟩ ↔ 𝐴 < 𝐶))
13 breq1 3968 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ < 𝐵𝐶 < 𝐵))
1412, 13orbi12d 783 . . 3 (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 <𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ < 𝐵) ↔ (𝐴 < 𝐶𝐶 < 𝐵)))
1514imbi2d 229 . 2 (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 < 𝐵 → (𝐴 <𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ < 𝐵)) ↔ (𝐴 < 𝐵 → (𝐴 < 𝐶𝐶 < 𝐵))))
16 ltsosr 7684 . . . 4 <R Or R
17 sowlin 4280 . . . 4 (( <R Or R ∧ (𝑥R𝑦R𝑧R)) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧𝑧 <R 𝑦)))
1816, 17mpan 421 . . 3 ((𝑥R𝑦R𝑧R) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧𝑧 <R 𝑦)))
19 ltresr 7759 . . 3 (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
20 ltresr 7759 . . . 4 (⟨𝑥, 0R⟩ <𝑧, 0R⟩ ↔ 𝑥 <R 𝑧)
21 ltresr 7759 . . . 4 (⟨𝑧, 0R⟩ <𝑦, 0R⟩ ↔ 𝑧 <R 𝑦)
2220, 21orbi12i 754 . . 3 ((⟨𝑥, 0R⟩ <𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <𝑦, 0R⟩) ↔ (𝑥 <R 𝑧𝑧 <R 𝑦))
2318, 19, 223imtr4g 204 . 2 ((𝑥R𝑦R𝑧R) → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ → (⟨𝑥, 0R⟩ <𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <𝑦, 0R⟩)))
241, 2, 3, 7, 11, 15, 233gencl 2746 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐴 < 𝐶𝐶 < 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 698  w3a 963   = wceq 1335  wcel 2128  cop 3563   class class class wbr 3965   Or wor 4255  Rcnr 7217  0Rc0r 7218   <R cltr 7223  cr 7731   < cltrr 7736
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-1o 6363  df-2o 6364  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-pli 7225  df-mi 7226  df-lti 7227  df-plpq 7264  df-mpq 7265  df-enq 7267  df-nqqs 7268  df-plqqs 7269  df-mqqs 7270  df-1nqqs 7271  df-rq 7272  df-ltnqqs 7273  df-enq0 7344  df-nq0 7345  df-0nq0 7346  df-plq0 7347  df-mq0 7348  df-inp 7386  df-i1p 7387  df-iplp 7388  df-iltp 7390  df-enr 7646  df-nr 7647  df-ltr 7650  df-0r 7651  df-r 7742  df-lt 7745
This theorem is referenced by: (None)
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