Theorem axpre-lttrn 11126

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 11253. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 11150. (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 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶))

Proof of Theorem axpre-lttrn

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elreal 11091	. 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 11091	. 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		elreal 11091	. 2 ⊢ (𝐶 ∈ ℝ ↔ ∃𝑧 ∈ R ⟨𝑧, 0R⟩ = 𝐶)
4		breq1 5113	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑦, 0R⟩))
5	4	anbi1d 631	. . 3 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩) ↔ (𝐴 <ℝ ⟨𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩)))
6		breq1 5113	. . 3 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑧, 0R⟩))
7	5, 6	imbi12d 344	. 2 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (((⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩) → ⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩) ↔ ((𝐴 <ℝ ⟨𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩) → 𝐴 <ℝ ⟨𝑧, 0R⟩)))
8		breq2 5114	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ 𝐵))
9		breq1 5113	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝐵 <ℝ ⟨𝑧, 0R⟩))
10	8, 9	anbi12d 632	. . 3 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <ℝ ⟨𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩) ↔ (𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ ⟨𝑧, 0R⟩)))
11	10	imbi1d 341	. 2 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (((𝐴 <ℝ ⟨𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩) → 𝐴 <ℝ ⟨𝑧, 0R⟩) ↔ ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ ⟨𝑧, 0R⟩) → 𝐴 <ℝ ⟨𝑧, 0R⟩)))
12		breq2 5114	. . . 4 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (𝐵 <ℝ ⟨𝑧, 0R⟩ ↔ 𝐵 <ℝ 𝐶))
13	12	anbi2d 630	. . 3 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ ⟨𝑧, 0R⟩) ↔ (𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶)))
14		breq2 5114	. . 3 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (𝐴 <ℝ ⟨𝑧, 0R⟩ ↔ 𝐴 <ℝ 𝐶))
15	13, 14	imbi12d 344	. 2 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ ⟨𝑧, 0R⟩) → 𝐴 <ℝ ⟨𝑧, 0R⟩) ↔ ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶)))
16		ltresr 11100	. . . . 5 ⊢ (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
17		ltresr 11100	. . . . 5 ⊢ (⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝑦 <R 𝑧)
18		ltsosr 11054	. . . . . 6 ⊢ <R Or R
19		ltrelsr 11028	. . . . . 6 ⊢ <R ⊆ (R × R)
20	18, 19	sotri 6103	. . . . 5 ⊢ ((𝑥 <R 𝑦 ∧ 𝑦 <R 𝑧) → 𝑥 <R 𝑧)
21	16, 17, 20	syl2anb 598	. . . 4 ⊢ ((⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩) → 𝑥 <R 𝑧)
22		ltresr 11100	. . . 4 ⊢ (⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝑥 <R 𝑧)
23	21, 22	sylibr 234	. . 3 ⊢ ((⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩) → ⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩)
24	23	a1i 11	. 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → ((⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩) → ⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩))
25	1, 2, 3, 7, 11, 15, 24	3gencl 3494	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ⟨cop 4598   class class class wbr 5110  Rcnr 10825  0_Rc0r 10826   <_R cltr 10831  ℝcr 11074   <_ℝ cltrr 11079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-oadd 8441  df-omul 8442  df-er 8674  df-ec 8676  df-qs 8680  df-ni 10832  df-pli 10833  df-mi 10834  df-lti 10835  df-plpq 10868  df-mpq 10869  df-ltpq 10870  df-enq 10871  df-nq 10872  df-erq 10873  df-plq 10874  df-mq 10875  df-1nq 10876  df-rq 10877  df-ltnq 10878  df-np 10941  df-1p 10942  df-plp 10943  df-ltp 10945  df-enr 11015  df-nr 11016  df-ltr 11019  df-0r 11020  df-r 11085  df-lt 11088

This theorem is referenced by: (None)