MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axpre-lttrn Structured version   Visualization version   GIF version

Theorem axpre-lttrn 11109
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 11234. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 11133. (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 11074 . 2 (𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)
2 elreal 11074 . 2 (𝐵 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐵)
3 elreal 11074 . 2 (𝐶 ∈ ℝ ↔ ∃𝑧R𝑧, 0R⟩ = 𝐶)
4 breq1 5113 . . . 4 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝐴 <𝑦, 0R⟩))
54anbi1d 631 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <𝑧, 0R⟩) ↔ (𝐴 <𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <𝑧, 0R⟩)))
6 breq1 5113 . . 3 (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <𝑧, 0R⟩ ↔ 𝐴 <𝑧, 0R⟩))
75, 6imbi12d 345 . 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 5114 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <𝑦, 0R⟩ ↔ 𝐴 < 𝐵))
9 breq1 5113 . . . 4 (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ <𝑧, 0R⟩ ↔ 𝐵 <𝑧, 0R⟩))
108, 9anbi12d 632 . . 3 (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <𝑧, 0R⟩) ↔ (𝐴 < 𝐵𝐵 <𝑧, 0R⟩)))
1110imbi1d 342 . 2 (⟨𝑦, 0R⟩ = 𝐵 → (((𝐴 <𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <𝑧, 0R⟩) → 𝐴 <𝑧, 0R⟩) ↔ ((𝐴 < 𝐵𝐵 <𝑧, 0R⟩) → 𝐴 <𝑧, 0R⟩)))
12 breq2 5114 . . . 4 (⟨𝑧, 0R⟩ = 𝐶 → (𝐵 <𝑧, 0R⟩ ↔ 𝐵 < 𝐶))
1312anbi2d 630 . . 3 (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 < 𝐵𝐵 <𝑧, 0R⟩) ↔ (𝐴 < 𝐵𝐵 < 𝐶)))
14 breq2 5114 . . 3 (⟨𝑧, 0R⟩ = 𝐶 → (𝐴 <𝑧, 0R⟩ ↔ 𝐴 < 𝐶))
1513, 14imbi12d 345 . 2 (⟨𝑧, 0R⟩ = 𝐶 → (((𝐴 < 𝐵𝐵 <𝑧, 0R⟩) → 𝐴 <𝑧, 0R⟩) ↔ ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶)))
16 ltresr 11083 . . . . 5 (⟨𝑥, 0R⟩ <𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
17 ltresr 11083 . . . . 5 (⟨𝑦, 0R⟩ <𝑧, 0R⟩ ↔ 𝑦 <R 𝑧)
18 ltsosr 11037 . . . . . 6 <R Or R
19 ltrelsr 11011 . . . . . 6 <R ⊆ (R × R)
2018, 19sotri 6086 . . . . 5 ((𝑥 <R 𝑦𝑦 <R 𝑧) → 𝑥 <R 𝑧)
2116, 17, 20syl2anb 599 . . . 4 ((⟨𝑥, 0R⟩ <𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <𝑧, 0R⟩) → 𝑥 <R 𝑧)
22 ltresr 11083 . . . 4 (⟨𝑥, 0R⟩ <𝑧, 0R⟩ ↔ 𝑥 <R 𝑧)
2321, 22sylibr 233 . . 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 3490 1 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵𝐵 < 𝐶) → 𝐴 < 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  cop 4597   class class class wbr 5110  Rcnr 10808  0Rc0r 10809   <R cltr 10814  cr 11057   < cltrr 11062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-oadd 8421  df-omul 8422  df-er 8655  df-ec 8657  df-qs 8661  df-ni 10815  df-pli 10816  df-mi 10817  df-lti 10818  df-plpq 10851  df-mpq 10852  df-ltpq 10853  df-enq 10854  df-nq 10855  df-erq 10856  df-plq 10857  df-mq 10858  df-1nq 10859  df-rq 10860  df-ltnq 10861  df-np 10924  df-1p 10925  df-plp 10926  df-ltp 10928  df-enr 10998  df-nr 10999  df-ltr 11002  df-0r 11003  df-r 11068  df-lt 11071
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator