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Theorem axpre-ltwlin 7845
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 7887. (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 7790 . 2 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
2 elreal 7790 . 2 (𝐵 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐵)
3 elreal 7790 . 2 (𝐶 ∈ ℝ ↔ ∃𝑧R𝑧, 0R⟩ = 𝐶)
4 breq1 3992 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝐴 <𝑦, 0R⟩))
5 breq1 3992 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑧, 0R⟩ ↔ 𝐴 <𝑧, 0R⟩))
65orbi1d 786 . . 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 3993 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <𝑦, 0R⟩ ↔ 𝐴 < 𝐵))
9 breq2 3993 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑧, 0R⟩ <𝑦, 0R⟩ ↔ ⟨𝑧, 0R⟩ < 𝐵))
109orbi2d 785 . . 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 3993 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → (𝐴 <𝑧, 0R⟩ ↔ 𝐴 < 𝐶))
13 breq1 3992 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ < 𝐵𝐶 < 𝐵))
1412, 13orbi12d 788 . . 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 7726 . . . 4 <R Or R
17 sowlin 4305 . . . 4 (( <R Or R ∧ (𝑥R𝑦R𝑧R)) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧𝑧 <R 𝑦)))
1816, 17mpan 422 . . 3 ((𝑥R𝑦R𝑧R) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧𝑧 <R 𝑦)))
19 ltresr 7801 . . 3 (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
20 ltresr 7801 . . . 4 (⟨𝑥, 0R⟩ <𝑧, 0R⟩ ↔ 𝑥 <R 𝑧)
21 ltresr 7801 . . . 4 (⟨𝑧, 0R⟩ <𝑦, 0R⟩ ↔ 𝑧 <R 𝑦)
2220, 21orbi12i 759 . . 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 2764 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐴 < 𝐶𝐶 < 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 703  w3a 973   = wceq 1348  wcel 2141  cop 3586   class class class wbr 3989   Or wor 4280  Rcnr 7259  0Rc0r 7260   <R cltr 7265  cr 7773   < cltrr 7778
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-i1p 7429  df-iplp 7430  df-iltp 7432  df-enr 7688  df-nr 7689  df-ltr 7692  df-0r 7693  df-r 7784  df-lt 7787
This theorem is referenced by: (None)
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