Theorem axpre-ltwlin 7049

Description: Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltwlin 7089. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.)

Assertion

Ref	Expression
axpre-ltwlin	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵)))

Proof of Theorem axpre-ltwlin

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

Step	Hyp	Ref	Expression
1		elreal 6997	. 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 6997	. 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		elreal 6997	. 2 ⊢ (𝐶 ∈ ℝ ↔ ∃𝑧 ∈ R ⟨𝑧, 0R⟩ = 𝐶)
4		breq1 3788	. . 3 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑦, 0R⟩))
5		breq1 3788	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑧, 0R⟩))
6	5	orbi1d 737	. . 3 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <ℝ ⟨𝑦, 0R⟩) ↔ (𝐴 <ℝ ⟨𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <ℝ ⟨𝑦, 0R⟩)))
7	4, 6	imbi12d 232	. 2 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ → (⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <ℝ ⟨𝑦, 0R⟩)) ↔ (𝐴 <ℝ ⟨𝑦, 0R⟩ → (𝐴 <ℝ ⟨𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <ℝ ⟨𝑦, 0R⟩))))
8		breq2 3789	. . 3 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ 𝐵))
9		breq2 3789	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑧, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ ⟨𝑧, 0R⟩ <ℝ 𝐵))
10	9	orbi2d 736	. . 3 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <ℝ ⟨𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <ℝ ⟨𝑦, 0R⟩) ↔ (𝐴 <ℝ ⟨𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <ℝ 𝐵)))
11	8, 10	imbi12d 232	. 2 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <ℝ ⟨𝑦, 0R⟩ → (𝐴 <ℝ ⟨𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <ℝ ⟨𝑦, 0R⟩)) ↔ (𝐴 <ℝ 𝐵 → (𝐴 <ℝ ⟨𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <ℝ 𝐵))))
12		breq2 3789	. . . 4 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (𝐴 <ℝ ⟨𝑧, 0R⟩ ↔ 𝐴 <ℝ 𝐶))
13		breq1 3788	. . . 4 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ <ℝ 𝐵 ↔ 𝐶 <ℝ 𝐵))
14	12, 13	orbi12d 739	. . 3 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 <ℝ ⟨𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <ℝ 𝐵) ↔ (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵)))
15	14	imbi2d 228	. 2 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 <ℝ 𝐵 → (𝐴 <ℝ ⟨𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <ℝ 𝐵)) ↔ (𝐴 <ℝ 𝐵 → (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵))))
16		ltsosr 6941	. . . 4 ⊢ <R Or R
17		sowlin 4075	. . . 4 ⊢ (( <R Or R ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R)) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦)))
18	16, 17	mpan 414	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦)))
19		ltresr 7007	. . 3 ⊢ (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
20		ltresr 7007	. . . 4 ⊢ (⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝑥 <R 𝑧)
21		ltresr 7007	. . . 4 ⊢ (⟨𝑧, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝑧 <R 𝑦)
22	20, 21	orbi12i 713	. . 3 ⊢ ((⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <ℝ ⟨𝑦, 0R⟩) ↔ (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦))
23	18, 19, 22	3imtr4g 203	. 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ → (⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <ℝ ⟨𝑦, 0R⟩)))
24	1, 2, 3, 7, 11, 15, 23	3gencl 2633	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵)))

Syntax hints:   → wi 4   ∨ wo 661   ∧ w3a 919   = wceq 1284   ∈ wcel 1433  ⟨cop 3401   class class class wbr 3785   Or wor 4050  Rcnr 6487  0_Rc0r 6488   <_R cltr 6493  ℝcr 6980   <_ℝ cltrr 6985

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-2o 6025  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-enq0 6614  df-nq0 6615  df-0nq0 6616  df-plq0 6617  df-mq0 6618  df-inp 6656  df-i1p 6657  df-iplp 6658  df-iltp 6660  df-enr 6903  df-nr 6904  df-ltr 6907  df-0r 6908  df-r 6991  df-lt 6994

This theorem is referenced by: (None)