Theorem axpre-lttrn 9866

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 9989. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 9890. (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 9831	. 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 9831	. 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		elreal 9831	. 2 ⊢ (𝐶 ∈ ℝ ↔ ∃𝑧 ∈ R ⟨𝑧, 0R⟩ = 𝐶)
4		breq1 4586	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑦, 0R⟩))
5	4	anbi1d 737	. . 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 4586	. . 3 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑧, 0R⟩))
7	5, 6	imbi12d 333	. 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 4587	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ 𝐵))
9		breq1 4586	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝐵 <ℝ ⟨𝑧, 0R⟩))
10	8, 9	anbi12d 743	. . 3 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <ℝ ⟨𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩) ↔ (𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ ⟨𝑧, 0R⟩)))
11	10	imbi1d 330	. 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 4587	. . . 4 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (𝐵 <ℝ ⟨𝑧, 0R⟩ ↔ 𝐵 <ℝ 𝐶))
13	12	anbi2d 736	. . 3 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ ⟨𝑧, 0R⟩) ↔ (𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶)))
14		breq2 4587	. . 3 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (𝐴 <ℝ ⟨𝑧, 0R⟩ ↔ 𝐴 <ℝ 𝐶))
15	13, 14	imbi12d 333	. 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 9840	. . . . 5 ⊢ (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
17		ltresr 9840	. . . . 5 ⊢ (⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝑦 <R 𝑧)
18		ltsosr 9794	. . . . . 6 ⊢ <R Or R
19		ltrelsr 9768	. . . . . 6 ⊢ <R ⊆ (R × R)
20	18, 19	sotri 5442	. . . . 5 ⊢ ((𝑥 <R 𝑦 ∧ 𝑦 <R 𝑧) → 𝑥 <R 𝑧)
21	16, 17, 20	syl2anb 495	. . . 4 ⊢ ((⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ∧ ⟨𝑦, 0R⟩ <ℝ ⟨𝑧, 0R⟩) → 𝑥 <R 𝑧)
22		ltresr 9840	. . . 4 ⊢ (⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝑥 <R 𝑧)
23	21, 22	sylibr 223	. . 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 3210	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   class class class wbr 4583  Rcnr 9566  0_Rc0r 9567   <_R cltr 9572  ℝcr 9814   <_ℝ cltrr 9819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-ec 7631  df-qs 7635  df-ni 9573  df-pli 9574  df-mi 9575  df-lti 9576  df-plpq 9609  df-mpq 9610  df-ltpq 9611  df-enq 9612  df-nq 9613  df-erq 9614  df-plq 9615  df-mq 9616  df-1nq 9617  df-rq 9618  df-ltnq 9619  df-np 9682  df-1p 9683  df-plp 9684  df-ltp 9686  df-enr 9756  df-nr 9757  df-ltr 9760  df-0r 9761  df-r 9825  df-lt 9828

This theorem is referenced by: (None)