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Theorem axpre-lttrn 10577
Description: Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axlttrn 10702. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 10601. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
axpre-lttrn ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))

Proof of Theorem axpre-lttrn
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elreal 10542 . 2 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
2 elreal 10542 . 2 (𝐵 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐵)
3 elreal 10542 . 2 (𝐶 ∈ ℝ ↔ ∃𝑧R𝑧, 0R⟩ = 𝐶)
4 breq1 5033 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝐴 <𝑦, 0R⟩))
54anbi1d 632 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <𝑧, 0R⟩) ↔ (𝐴 <𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <𝑧, 0R⟩)))
6 breq1 5033 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑧, 0R⟩ ↔ 𝐴 <𝑧, 0R⟩))
75, 6imbi12d 348 . 2 (⟨𝑥, 0R⟩ = 𝐴 → (((⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <𝑧, 0R⟩) → ⟨𝑥, 0R⟩ <𝑧, 0R⟩) ↔ ((𝐴 <𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <𝑧, 0R⟩) → 𝐴 <𝑧, 0R⟩)))
8 breq2 5034 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <𝑦, 0R⟩ ↔ 𝐴 < 𝐵))
9 breq1 5033 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ <𝑧, 0R⟩ ↔ 𝐵 <𝑧, 0R⟩))
108, 9anbi12d 633 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <𝑧, 0R⟩) ↔ (𝐴 < 𝐵𝐵 <𝑧, 0R⟩)))
1110imbi1d 345 . 2 (⟨𝑦, 0R⟩ = 𝐵 → (((𝐴 <𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <𝑧, 0R⟩) → 𝐴 <𝑧, 0R⟩) ↔ ((𝐴 < 𝐵𝐵 <𝑧, 0R⟩) → 𝐴 <𝑧, 0R⟩)))
12 breq2 5034 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → (𝐵 <𝑧, 0R⟩ ↔ 𝐵 < 𝐶))
1312anbi2d 631 . . 3 (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 < 𝐵𝐵 <𝑧, 0R⟩) ↔ (𝐴 < 𝐵𝐵 < 𝐶)))
14 breq2 5034 . . 3 (⟨𝑧, 0R⟩ = 𝐶 → (𝐴 <𝑧, 0R⟩ ↔ 𝐴 < 𝐶))
1513, 14imbi12d 348 . 2 (⟨𝑧, 0R⟩ = 𝐶 → (((𝐴 < 𝐵𝐵 <𝑧, 0R⟩) → 𝐴 <𝑧, 0R⟩) ↔ ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶)))
16 ltresr 10551 . . . . 5 (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
17 ltresr 10551 . . . . 5 (⟨𝑦, 0R⟩ <𝑧, 0R⟩ ↔ 𝑦 <R 𝑧)
18 ltsosr 10505 . . . . . 6 <R Or R
19 ltrelsr 10479 . . . . . 6 <R ⊆ (R × R)
2018, 19sotri 5954 . . . . 5 ((𝑥 <R 𝑦𝑦 <R 𝑧) → 𝑥 <R 𝑧)
2116, 17, 20syl2anb 600 . . . 4 ((⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <𝑧, 0R⟩) → 𝑥 <R 𝑧)
22 ltresr 10551 . . . 4 (⟨𝑥, 0R⟩ <𝑧, 0R⟩ ↔ 𝑥 <R 𝑧)
2321, 22sylibr 237 . . 3 ((⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <𝑧, 0R⟩) → ⟨𝑥, 0R⟩ <𝑧, 0R⟩)
2423a1i 11 . 2 ((𝑥R𝑦R𝑧R) → ((⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <𝑧, 0R⟩) → ⟨𝑥, 0R⟩ <𝑧, 0R⟩))
251, 2, 3, 7, 11, 15, 243gencl 3483 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2111  cop 4531   class class class wbr 5030  Rcnr 10276  0Rc0r 10277   <R cltr 10282  cr 10525   < cltrr 10530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-omul 8090  df-er 8272  df-ec 8274  df-qs 8278  df-ni 10283  df-pli 10284  df-mi 10285  df-lti 10286  df-plpq 10319  df-mpq 10320  df-ltpq 10321  df-enq 10322  df-nq 10323  df-erq 10324  df-plq 10325  df-mq 10326  df-1nq 10327  df-rq 10328  df-ltnq 10329  df-np 10392  df-1p 10393  df-plp 10394  df-ltp 10396  df-enr 10466  df-nr 10467  df-ltr 10470  df-0r 10471  df-r 10536  df-lt 10539
This theorem is referenced by: (None)
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