Theorem axpre-lttri 11186

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 11313. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 11210. (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 11152	. 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 11152	. 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		breq1 5146	. . 3 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑦, 0R⟩))
4		eqeq1 2729	. . . . 5 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ 𝐴 = ⟨𝑦, 0R⟩))
5		breq2 5147	. . . . 5 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩ ↔ ⟨𝑦, 0R⟩ <ℝ 𝐴))
6	4, 5	orbi12d 916	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩) ↔ (𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ 𝐴)))
7	6	notbid 317	. . 3 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (¬ (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩) ↔ ¬ (𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ 𝐴)))
8	3, 7	bibi12d 344	. 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 5147	. . 3 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ 𝐵))
10		eqeq2 2737	. . . . 5 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 = ⟨𝑦, 0R⟩ ↔ 𝐴 = 𝐵))
11		breq1 5146	. . . . 5 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ <ℝ 𝐴 ↔ 𝐵 <ℝ 𝐴))
12	10, 11	orbi12d 916	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ 𝐴) ↔ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴)))
13	12	notbid 317	. . 3 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (¬ (𝐴 = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ 𝐴) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴)))
14	9, 13	bibi12d 344	. 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 11115	. . . 4 ⊢ <R Or R
16		sotric 5612	. . . 4 ⊢ (( <R Or R ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) → (𝑥 <R 𝑦 ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦 <R 𝑥)))
17	15, 16	mpan 688	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 <R 𝑦 ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦 <R 𝑥)))
18		ltresr 11161	. . 3 ⊢ (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
19		vex 3467	. . . . . 6 ⊢ 𝑥 ∈ V
20	19	eqresr 11158	. . . . 5 ⊢ (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ 𝑥 = 𝑦)
21		ltresr 11161	. . . . 5 ⊢ (⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩ ↔ 𝑦 <R 𝑥)
22	20, 21	orbi12i 912	. . . 4 ⊢ ((⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩) ↔ (𝑥 = 𝑦 ∨ 𝑦 <R 𝑥))
23	22	notbii 319	. . 3 ⊢ (¬ (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩) ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦 <R 𝑥))
24	17, 18, 23	3bitr4g 313	. 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 3507	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   = wceq 1533   ∈ wcel 2098  ⟨cop 4630   class class class wbr 5143   Or wor 5583  Rcnr 10886  0_Rc0r 10887   <_R cltr 10892  ℝcr 11135   <_ℝ cltrr 11140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-inf2 9662

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-oadd 8487  df-omul 8488  df-er 8721  df-ec 8723  df-qs 8727  df-ni 10893  df-pli 10894  df-mi 10895  df-lti 10896  df-plpq 10929  df-mpq 10930  df-ltpq 10931  df-enq 10932  df-nq 10933  df-erq 10934  df-plq 10935  df-mq 10936  df-1nq 10937  df-rq 10938  df-ltnq 10939  df-np 11002  df-1p 11003  df-plp 11004  df-ltp 11006  df-enr 11076  df-nr 11077  df-ltr 11080  df-0r 11081  df-r 11146  df-lt 11149

This theorem is referenced by: (None)