Theorem axpre-lttri 9930

Description: Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axlttri 10053. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 9954. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
axpre-lttri	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴)))

Proof of Theorem axpre-lttri

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

Step	Hyp	Ref	Expression
1		elreal 9896	. 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 9896	. 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		breq1 4616	. . 3 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑦, 0R⟩))
4		eqeq1 2625	. . . . 5 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ 𝐴 = ⟨𝑦, 0R⟩))
5		breq2 4617	. . . . 5 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩ ↔ ⟨𝑦, 0R⟩ <ℝ 𝐴))
6	4, 5	orbi12d 745	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩) ↔ (𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ 𝐴)))
7	6	notbid 308	. . 3 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (¬ (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩) ↔ ¬ (𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ 𝐴)))
8	3, 7	bibi12d 335	. 2 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ ¬ (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩)) ↔ (𝐴 <ℝ ⟨𝑦, 0R⟩ ↔ ¬ (𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ 𝐴))))
9		breq2 4617	. . 3 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ 𝐵))
10		eqeq2 2632	. . . . 5 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 = ⟨𝑦, 0R⟩ ↔ 𝐴 = 𝐵))
11		breq1 4616	. . . . 5 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ <ℝ 𝐴 ↔ 𝐵 <ℝ 𝐴))
12	10, 11	orbi12d 745	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ 𝐴) ↔ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴)))
13	12	notbid 308	. . 3 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (¬ (𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ 𝐴) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴)))
14	9, 13	bibi12d 335	. 2 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <ℝ ⟨𝑦, 0R⟩ ↔ ¬ (𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ 𝐴)) ↔ (𝐴 <ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴))))
15		ltsosr 9859	. . . 4 ⊢ <R Or R
16		sotric 5021	. . . 4 ⊢ (( <R Or R ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) → (𝑥 <R 𝑦 ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦 <R 𝑥)))
17	15, 16	mpan 705	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 <R 𝑦 ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦 <R 𝑥)))
18		ltresr 9905	. . 3 ⊢ (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
19		vex 3189	. . . . . 6 ⊢ 𝑥 ∈ V
20	19	eqresr 9902	. . . . 5 ⊢ (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ 𝑥 = 𝑦)
21		ltresr 9905	. . . . 5 ⊢ (⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩ ↔ 𝑦 <R 𝑥)
22	20, 21	orbi12i 543	. . . 4 ⊢ ((⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩) ↔ (𝑥 = 𝑦 ∨ 𝑦 <R 𝑥))
23	22	notbii 310	. . 3 ⊢ (¬ (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩) ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦 <R 𝑥))
24	17, 18, 23	3bitr4g 303	. 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ ¬ (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩)))
25	1, 2, 8, 14, 24	2gencl 3222	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ⟨cop 4154   class class class wbr 4613   Or wor 4994  Rcnr 9631  0_Rc0r 9632   <_R cltr 9637  ℝcr 9879   <_ℝ 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)