Theorem axpre-apti 8018

Description: Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-apti 8060.

(Contributed by Jim Kingdon, 29-Jan-2020.) (New usage is discouraged.)

Assertion

Ref	Expression
axpre-apti	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)) → 𝐴 = 𝐵)

Proof of Theorem axpre-apti

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

Step	Hyp	Ref	Expression
1		elreal 7961	. . 3 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R ⟨𝑥, 0R⟩ = 𝐴)
2		elreal 7961	. . 3 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐵)
3		breq1 4054	. . . . . 6 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ ⟨𝑦, 0R⟩))
4		breq2 4055	. . . . . 6 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩ ↔ ⟨𝑦, 0R⟩ <ℝ 𝐴))
5	3, 4	orbi12d 795	. . . . 5 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → ((⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩) ↔ (𝐴 <ℝ ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ 𝐴)))
6	5	notbid 669	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (¬ (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩) ↔ ¬ (𝐴 <ℝ ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ 𝐴)))
7		eqeq1 2213	. . . 4 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ 𝐴 = ⟨𝑦, 0R⟩))
8	6, 7	imbi12d 234	. . 3 ⊢ (⟨𝑥, 0R⟩ = 𝐴 → ((¬ (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩) → ⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩) ↔ (¬ (𝐴 <ℝ ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ 𝐴) → 𝐴 = ⟨𝑦, 0R⟩)))
9		breq2 4055	. . . . . 6 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 <ℝ ⟨𝑦, 0R⟩ ↔ 𝐴 <ℝ 𝐵))
10		breq1 4054	. . . . . 6 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (⟨𝑦, 0R⟩ <ℝ 𝐴 ↔ 𝐵 <ℝ 𝐴))
11	9, 10	orbi12d 795	. . . . 5 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → ((𝐴 <ℝ ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ 𝐴) ↔ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)))
12	11	notbid 669	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (¬ (𝐴 <ℝ ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ 𝐴) ↔ ¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)))
13		eqeq2 2216	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → (𝐴 = ⟨𝑦, 0R⟩ ↔ 𝐴 = 𝐵))
14	12, 13	imbi12d 234	. . 3 ⊢ (⟨𝑦, 0R⟩ = 𝐵 → ((¬ (𝐴 <ℝ ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ 𝐴) → 𝐴 = ⟨𝑦, 0R⟩) ↔ (¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴) → 𝐴 = 𝐵)))
15		aptisr 7912	. . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ ¬ (𝑥 <R 𝑦 ∨ 𝑦 <R 𝑥)) → 𝑥 = 𝑦)
16	15	3expia 1208	. . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (¬ (𝑥 <R 𝑦 ∨ 𝑦 <R 𝑥) → 𝑥 = 𝑦))
17		ltresr 7972	. . . . . 6 ⊢ (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 𝑥 <R 𝑦)
18		ltresr 7972	. . . . . 6 ⊢ (⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩ ↔ 𝑦 <R 𝑥)
19	17, 18	orbi12i 766	. . . . 5 ⊢ ((⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩) ↔ (𝑥 <R 𝑦 ∨ 𝑦 <R 𝑥))
20	19	notbii 670	. . . 4 ⊢ (¬ (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩) ↔ ¬ (𝑥 <R 𝑦 ∨ 𝑦 <R 𝑥))
21		vex 2776	. . . . 5 ⊢ 𝑥 ∈ V
22	21	eqresr 7969	. . . 4 ⊢ (⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩ ↔ 𝑥 = 𝑦)
23	16, 20, 22	3imtr4g 205	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (¬ (⟨𝑥, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ∨ ⟨𝑦, 0R⟩ <ℝ ⟨𝑥, 0R⟩) → ⟨𝑥, 0R⟩ = ⟨𝑦, 0R⟩))
24	1, 2, 8, 14, 23	2gencl 2807	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴) → 𝐴 = 𝐵))
25	24	3impia 1203	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)) → 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 710   ∧ w3a 981   = wceq 1373   ∈ wcel 2177  ⟨cop 3641   class class class wbr 4051  Rcnr 7430  0_Rc0r 7431   <_R cltr 7436  ℝcr 7944   <_ℝ cltrr 7949

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-eprel 4344  df-id 4348  df-po 4351  df-iso 4352  df-iord 4421  df-on 4423  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-irdg 6469  df-1o 6515  df-2o 6516  df-oadd 6519  df-omul 6520  df-er 6633  df-ec 6635  df-qs 6639  df-ni 7437  df-pli 7438  df-mi 7439  df-lti 7440  df-plpq 7477  df-mpq 7478  df-enq 7480  df-nqqs 7481  df-plqqs 7482  df-mqqs 7483  df-1nqqs 7484  df-rq 7485  df-ltnqqs 7486  df-enq0 7557  df-nq0 7558  df-0nq0 7559  df-plq0 7560  df-mq0 7561  df-inp 7599  df-i1p 7600  df-iplp 7601  df-iltp 7603  df-enr 7859  df-nr 7860  df-ltr 7863  df-0r 7864  df-r 7955  df-lt 7958

This theorem is referenced by: (None)