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Theorem axpre-ltwlin 7885
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 7927. (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 7830 . 2 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
2 elreal 7830 . 2 (𝐵 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐵)
3 elreal 7830 . 2 (𝐶 ∈ ℝ ↔ ∃𝑧R𝑧, 0R⟩ = 𝐶)
4 breq1 4008 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝐴 <𝑦, 0R⟩))
5 breq1 4008 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑧, 0R⟩ ↔ 𝐴 <𝑧, 0R⟩))
65orbi1d 791 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ <𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <𝑦, 0R⟩) ↔ (𝐴 <𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <𝑦, 0R⟩)))
74, 6imbi12d 234 . 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 4009 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <𝑦, 0R⟩ ↔ 𝐴 < 𝐵))
9 breq2 4009 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑧, 0R⟩ <𝑦, 0R⟩ ↔ ⟨𝑧, 0R⟩ < 𝐵))
109orbi2d 790 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <𝑦, 0R⟩) ↔ (𝐴 <𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ < 𝐵)))
118, 10imbi12d 234 . 2 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <𝑦, 0R⟩ → (𝐴 <𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <𝑦, 0R⟩)) ↔ (𝐴 < 𝐵 → (𝐴 <𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ < 𝐵))))
12 breq2 4009 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → (𝐴 <𝑧, 0R⟩ ↔ 𝐴 < 𝐶))
13 breq1 4008 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ < 𝐵𝐶 < 𝐵))
1412, 13orbi12d 793 . . 3 (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 <𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ < 𝐵) ↔ (𝐴 < 𝐶𝐶 < 𝐵)))
1514imbi2d 230 . 2 (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 < 𝐵 → (𝐴 <𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ < 𝐵)) ↔ (𝐴 < 𝐵 → (𝐴 < 𝐶𝐶 < 𝐵))))
16 ltsosr 7766 . . . 4 <R Or R
17 sowlin 4322 . . . 4 (( <R Or R ∧ (𝑥R𝑦R𝑧R)) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧𝑧 <R 𝑦)))
1816, 17mpan 424 . . 3 ((𝑥R𝑦R𝑧R) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧𝑧 <R 𝑦)))
19 ltresr 7841 . . 3 (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
20 ltresr 7841 . . . 4 (⟨𝑥, 0R⟩ <𝑧, 0R⟩ ↔ 𝑥 <R 𝑧)
21 ltresr 7841 . . . 4 (⟨𝑧, 0R⟩ <𝑦, 0R⟩ ↔ 𝑧 <R 𝑦)
2220, 21orbi12i 764 . . 3 ((⟨𝑥, 0R⟩ <𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <𝑦, 0R⟩) ↔ (𝑥 <R 𝑧𝑧 <R 𝑦))
2318, 19, 223imtr4g 205 . 2 ((𝑥R𝑦R𝑧R) → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ → (⟨𝑥, 0R⟩ <𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <𝑦, 0R⟩)))
241, 2, 3, 7, 11, 15, 233gencl 2773 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐴 < 𝐶𝐶 < 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 708  w3a 978   = wceq 1353  wcel 2148  cop 3597   class class class wbr 4005   Or wor 4297  Rcnr 7299  0Rc0r 7300   <R cltr 7305  cr 7813   < cltrr 7818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-irdg 6374  df-1o 6420  df-2o 6421  df-oadd 6424  df-omul 6425  df-er 6538  df-ec 6540  df-qs 6544  df-ni 7306  df-pli 7307  df-mi 7308  df-lti 7309  df-plpq 7346  df-mpq 7347  df-enq 7349  df-nqqs 7350  df-plqqs 7351  df-mqqs 7352  df-1nqqs 7353  df-rq 7354  df-ltnqqs 7355  df-enq0 7426  df-nq0 7427  df-0nq0 7428  df-plq0 7429  df-mq0 7430  df-inp 7468  df-i1p 7469  df-iplp 7470  df-iltp 7472  df-enr 7728  df-nr 7729  df-ltr 7732  df-0r 7733  df-r 7824  df-lt 7827
This theorem is referenced by: (None)
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