Theorem axpre-ltwlin 8240

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 8282. (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 8185	. 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 8185	. 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		elreal 8185	. 2 ⊢ (𝐶 ∈ ℝ ↔ ∃𝑧 ∈ R ⟨𝑧, 0R⟩ = 𝐶)
4		breq1 4128	. . 3 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑦, 0R⟩))
5		breq1 4128	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑧, 0R⟩))
6	5	orbi1d 803	. . 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 4129	. . 3 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ 𝐵))
9		breq2 4129	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑧, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ ⟨𝑧, 0R⟩ <ℝ 𝐵))
10	9	orbi2d 802	. . 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 4129	. . . 4 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (𝐴 <ℝ ⟨𝑧, 0R⟩ ↔ 𝐴 <ℝ 𝐶))
13		breq1 4128	. . . 4 ⊢ (⟨𝑧, 0R⟩ = 𝐶 → (⟨𝑧, 0R⟩ <ℝ 𝐵 ↔ 𝐶 <ℝ 𝐵))
14	12, 13	orbi12d 805	. . 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 8121	. . . 4 ⊢ <R Or R
17		sowlin 4460	. . . 4 ⊢ (( <R Or R ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R)) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦)))
18	16, 17	mpan 428	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦)))
19		ltresr 8196	. . 3 ⊢ (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
20		ltresr 8196	. . . 4 ⊢ (⟨𝑥, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 𝑥 <R 𝑧)
21		ltresr 8196	. . . 4 ⊢ (⟨𝑧, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝑧 <R 𝑦)
22	20, 21	orbi12i 776	. . 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 2856	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵)))

Syntax hints:   → wi 4   ∨ wo 720   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ⟨cop 3708   class class class wbr 4125   Or wor 4435  Rcnr 7654  0_Rc0r 7655   <_R cltr 7660  ℝcr 8168   <_ℝ cltrr 8173

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-2o 6678  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-enq0 7781  df-nq0 7782  df-0nq0 7783  df-plq0 7784  df-mq0 7785  df-inp 7823  df-i1p 7824  df-iplp 7825  df-iltp 7827  df-enr 8083  df-nr 8084  df-ltr 8087  df-0r 8088  df-r 8179  df-lt 8182

This theorem is referenced by: (None)