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Theorem axpre-lttrn 9931
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 10054. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 9955. (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 9896 . 2 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
2 elreal 9896 . 2 (𝐵 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐵)
3 elreal 9896 . 2 (𝐶 ∈ ℝ ↔ ∃𝑧R𝑧, 0R⟩ = 𝐶)
4 breq1 4616 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝐴 <𝑦, 0R⟩))
54anbi1d 740 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <𝑧, 0R⟩) ↔ (𝐴 <𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <𝑧, 0R⟩)))
6 breq1 4616 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑧, 0R⟩ ↔ 𝐴 <𝑧, 0R⟩))
75, 6imbi12d 334 . 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 4617 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <𝑦, 0R⟩ ↔ 𝐴 < 𝐵))
9 breq1 4616 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ <𝑧, 0R⟩ ↔ 𝐵 <𝑧, 0R⟩))
108, 9anbi12d 746 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <𝑧, 0R⟩) ↔ (𝐴 < 𝐵𝐵 <𝑧, 0R⟩)))
1110imbi1d 331 . 2 (⟨𝑦, 0R⟩ = 𝐵 → (((𝐴 <𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <𝑧, 0R⟩) → 𝐴 <𝑧, 0R⟩) ↔ ((𝐴 < 𝐵𝐵 <𝑧, 0R⟩) → 𝐴 <𝑧, 0R⟩)))
12 breq2 4617 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → (𝐵 <𝑧, 0R⟩ ↔ 𝐵 < 𝐶))
1312anbi2d 739 . . 3 (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 < 𝐵𝐵 <𝑧, 0R⟩) ↔ (𝐴 < 𝐵𝐵 < 𝐶)))
14 breq2 4617 . . 3 (⟨𝑧, 0R⟩ = 𝐶 → (𝐴 <𝑧, 0R⟩ ↔ 𝐴 < 𝐶))
1513, 14imbi12d 334 . 2 (⟨𝑧, 0R⟩ = 𝐶 → (((𝐴 < 𝐵𝐵 <𝑧, 0R⟩) → 𝐴 <𝑧, 0R⟩) ↔ ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶)))
16 ltresr 9905 . . . . 5 (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
17 ltresr 9905 . . . . 5 (⟨𝑦, 0R⟩ <𝑧, 0R⟩ ↔ 𝑦 <R 𝑧)
18 ltsosr 9859 . . . . . 6 <R Or R
19 ltrelsr 9833 . . . . . 6 <R ⊆ (R × R)
2018, 19sotri 5482 . . . . 5 ((𝑥 <R 𝑦𝑦 <R 𝑧) → 𝑥 <R 𝑧)
2116, 17, 20syl2anb 496 . . . 4 ((⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <𝑧, 0R⟩) → 𝑥 <R 𝑧)
22 ltresr 9905 . . . 4 (⟨𝑥, 0R⟩ <𝑧, 0R⟩ ↔ 𝑥 <R 𝑧)
2321, 22sylibr 224 . . 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 3223 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  cop 4154   class class class wbr 4613  Rcnr 9631  0Rc0r 9632   <R cltr 9637  cr 9879   < cltrr 9884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-omul 7510  df-er 7687  df-ec 7689  df-qs 7693  df-ni 9638  df-pli 9639  df-mi 9640  df-lti 9641  df-plpq 9674  df-mpq 9675  df-ltpq 9676  df-enq 9677  df-nq 9678  df-erq 9679  df-plq 9680  df-mq 9681  df-1nq 9682  df-rq 9683  df-ltnq 9684  df-np 9747  df-1p 9748  df-plp 9749  df-ltp 9751  df-enr 9821  df-nr 9822  df-ltr 9825  df-0r 9826  df-r 9890  df-lt 9893
This theorem is referenced by: (None)
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