Theorem axpre-ltwlin 7803

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 7845. (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 7748	. 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 7748	. 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		elreal 7748	. 2 ⊢ (𝐶 ∈ ℝ ↔ ∃𝑧 ∈ R ⟨𝑧, 0R⟩ = 𝐶)
4		breq1 3968	. . 3 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑦, 0R⟩))
5		breq1 3968	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑧, 0R⟩))
6	5	orbi1d 781	. . 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 233	. 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 3969	. . 3 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ 𝐵))
9		breq2 3969	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑧, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ ⟨𝑧, 0R⟩ <ℝ 𝐵))
10	9	orbi2d 780	. . 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 233	. 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 3969	. . . 4 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (𝐴 <ℝ ⟨𝑧, 0R⟩ ↔ 𝐴 <ℝ 𝐶))
13		breq1 3968	. . . 4 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ <ℝ 𝐵 ↔ 𝐶 <ℝ 𝐵))
14	12, 13	orbi12d 783	. . 3 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 <ℝ ⟨𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <ℝ 𝐵) ↔ (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵)))
15	14	imbi2d 229	. 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 7684	. . . 4 ⊢ <R Or R
17		sowlin 4280	. . . 4 ⊢ (( <R Or R ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R)) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦)))
18	16, 17	mpan 421	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦)))
19		ltresr 7759	. . 3 ⊢ (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
20		ltresr 7759	. . . 4 ⊢ (⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝑥 <R 𝑧)
21		ltresr 7759	. . . 4 ⊢ (⟨𝑧, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝑧 <R 𝑦)
22	20, 21	orbi12i 754	. . 3 ⊢ ((⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <ℝ ⟨𝑦, 0R⟩) ↔ (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦))
23	18, 19, 22	3imtr4g 204	. 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 2746	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵)))

Syntax hints:   → wi 4   ∨ wo 698   ∧ w3a 963   = wceq 1335   ∈ wcel 2128  ⟨cop 3563   class class class wbr 3965   Or wor 4255  Rcnr 7217  0_Rc0r 7218   <_R cltr 7223  ℝcr 7731   <_ℝ cltrr 7736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-1o 6363  df-2o 6364  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-pli 7225  df-mi 7226  df-lti 7227  df-plpq 7264  df-mpq 7265  df-enq 7267  df-nqqs 7268  df-plqqs 7269  df-mqqs 7270  df-1nqqs 7271  df-rq 7272  df-ltnqqs 7273  df-enq0 7344  df-nq0 7345  df-0nq0 7346  df-plq0 7347  df-mq0 7348  df-inp 7386  df-i1p 7387  df-iplp 7388  df-iltp 7390  df-enr 7646  df-nr 7647  df-ltr 7650  df-0r 7651  df-r 7742  df-lt 7745

This theorem is referenced by: (None)