Theorem axpre-ltwlin 8081

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 8123. (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 8026	. 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 8026	. 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		elreal 8026	. 2 ⊢ (𝐶 ∈ ℝ ↔ ∃𝑧 ∈ R ⟨𝑧, 0R⟩ = 𝐶)
4		breq1 4086	. . 3 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑦, 0R⟩))
5		breq1 4086	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑧, 0R⟩))
6	5	orbi1d 796	. . 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 234	. 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 4087	. . 3 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ 𝐵))
9		breq2 4087	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑧, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ ⟨𝑧, 0R⟩ <ℝ 𝐵))
10	9	orbi2d 795	. . 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 234	. 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 4087	. . . 4 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (𝐴 <ℝ ⟨𝑧, 0R⟩ ↔ 𝐴 <ℝ 𝐶))
13		breq1 4086	. . . 4 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ <ℝ 𝐵 ↔ 𝐶 <ℝ 𝐵))
14	12, 13	orbi12d 798	. . 3 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → ((𝐴 <ℝ ⟨𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <ℝ 𝐵) ↔ (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵)))
15	14	imbi2d 230	. 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 7962	. . . 4 ⊢ <R Or R
17		sowlin 4411	. . . 4 ⊢ (( <R Or R ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R)) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦)))
18	16, 17	mpan 424	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦)))
19		ltresr 8037	. . 3 ⊢ (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
20		ltresr 8037	. . . 4 ⊢ (⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝑥 <R 𝑧)
21		ltresr 8037	. . . 4 ⊢ (⟨𝑧, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝑧 <R 𝑦)
22	20, 21	orbi12i 769	. . 3 ⊢ ((⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ∨ ⟨𝑧, 0R⟩ <ℝ ⟨𝑦, 0R⟩) ↔ (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦))
23	18, 19, 22	3imtr4g 205	. 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 2834	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵)))

Syntax hints:   → wi 4   ∨ wo 713   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ⟨cop 3669   class class class wbr 4083   Or wor 4386  Rcnr 7495  0_Rc0r 7496   <_R cltr 7501  ℝcr 8009   <_ℝ cltrr 8014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-1o 6568  df-2o 6569  df-oadd 6572  df-omul 6573  df-er 6688  df-ec 6690  df-qs 6694  df-ni 7502  df-pli 7503  df-mi 7504  df-lti 7505  df-plpq 7542  df-mpq 7543  df-enq 7545  df-nqqs 7546  df-plqqs 7547  df-mqqs 7548  df-1nqqs 7549  df-rq 7550  df-ltnqqs 7551  df-enq0 7622  df-nq0 7623  df-0nq0 7624  df-plq0 7625  df-mq0 7626  df-inp 7664  df-i1p 7665  df-iplp 7666  df-iltp 7668  df-enr 7924  df-nr 7925  df-ltr 7928  df-0r 7929  df-r 8020  df-lt 8023

This theorem is referenced by: (None)