ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  axpre-ltwlin GIF version

Theorem axpre-ltwlin 7698
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 7740. (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 7643 . 2 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
2 elreal 7643 . 2 (𝐵 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐵)
3 elreal 7643 . 2 (𝐶 ∈ ℝ ↔ ∃𝑧R𝑧, 0R⟩ = 𝐶)
4 breq1 3932 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝐴 <𝑦, 0R⟩))
5 breq1 3932 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑧, 0R⟩ ↔ 𝐴 <𝑧, 0R⟩))
65orbi1d 780 . . 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 3933 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <𝑦, 0R⟩ ↔ 𝐴 < 𝐵))
9 breq2 3933 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑧, 0R⟩ <𝑦, 0R⟩ ↔ ⟨𝑧, 0R⟩ < 𝐵))
109orbi2d 779 . . 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 3933 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → (𝐴 <𝑧, 0R⟩ ↔ 𝐴 < 𝐶))
13 breq1 3932 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ < 𝐵𝐶 < 𝐵))
1412, 13orbi12d 782 . . 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 7579 . . . 4 <R Or R
17 sowlin 4242 . . . 4 (( <R Or R ∧ (𝑥R𝑦R𝑧R)) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧𝑧 <R 𝑦)))
1816, 17mpan 420 . . 3 ((𝑥R𝑦R𝑧R) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧𝑧 <R 𝑦)))
19 ltresr 7654 . . 3 (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
20 ltresr 7654 . . . 4 (⟨𝑥, 0R⟩ <𝑧, 0R⟩ ↔ 𝑥 <R 𝑧)
21 ltresr 7654 . . . 4 (⟨𝑧, 0R⟩ <𝑦, 0R⟩ ↔ 𝑧 <R 𝑦)
2220, 21orbi12i 753 . . 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 2720 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐴 < 𝐶𝐶 < 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 697  w3a 962   = wceq 1331  wcel 1480  cop 3530   class class class wbr 3929   Or wor 4217  Rcnr 7112  0Rc0r 7113   <R cltr 7118  cr 7626   < cltrr 7631
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-mqqs 7165  df-1nqqs 7166  df-rq 7167  df-ltnqqs 7168  df-enq0 7239  df-nq0 7240  df-0nq0 7241  df-plq0 7242  df-mq0 7243  df-inp 7281  df-i1p 7282  df-iplp 7283  df-iltp 7285  df-enr 7541  df-nr 7542  df-ltr 7545  df-0r 7546  df-r 7637  df-lt 7640
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator