Theorem axpre-apti 6749

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 6778.

(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 6707	. . 3 ⊢ (A ∈ ℝ ↔ ∃x ∈ R ⟨x, 0R⟩ = A)
2		elreal 6707	. . 3 ⊢ (B ∈ ℝ ↔ ∃y ∈ R ⟨y, 0R⟩ = B)
3		breq1 3758	. . . . . 6 ⊢ (⟨x, 0R⟩ = A → (⟨x, 0R⟩ <ℝ ⟨y, 0R⟩ ↔ A <ℝ ⟨y, 0R⟩))
4		breq2 3759	. . . . . 6 ⊢ (⟨x, 0R⟩ = A → (⟨y, 0R⟩ <ℝ ⟨x, 0R⟩ ↔ ⟨y, 0R⟩ <ℝ A))
5	3, 4	orbi12d 706	. . . . 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 591	. . . 4 ⊢ (⟨x, 0R⟩ = A → (¬ (⟨x, 0R⟩ <ℝ ⟨y, 0R⟩ ∨ ⟨y, 0R⟩ <ℝ ⟨x, 0R⟩) ↔ ¬ (A <ℝ ⟨y, 0R⟩ ∨ ⟨y, 0R⟩ <ℝ A)))
7		eqeq1 2043	. . . 4 ⊢ (⟨x, 0R⟩ = A → (⟨x, 0R⟩ = ⟨y, 0R⟩ ↔ A = ⟨y, 0R⟩))
8	6, 7	imbi12d 223	. . 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 3759	. . . . . 6 ⊢ (⟨y, 0R⟩ = B → (A <ℝ ⟨y, 0R⟩ ↔ A <ℝ B))
10		breq1 3758	. . . . . 6 ⊢ (⟨y, 0R⟩ = B → (⟨y, 0R⟩ <ℝ A ↔ B <ℝ A))
11	9, 10	orbi12d 706	. . . . 5 ⊢ (⟨y, 0R⟩ = B → ((A <ℝ ⟨y, 0R⟩ ∨ ⟨y, 0R⟩ <ℝ A) ↔ (A <ℝ B ∨ B <ℝ A)))
12	11	notbid 591	. . . 4 ⊢ (⟨y, 0R⟩ = B → (¬ (A <ℝ ⟨y, 0R⟩ ∨ ⟨y, 0R⟩ <ℝ A) ↔ ¬ (A <ℝ B ∨ B <ℝ A)))
13		eqeq2 2046	. . . 4 ⊢ (⟨y, 0R⟩ = B → (A = ⟨y, 0R⟩ ↔ A = B))
14	12, 13	imbi12d 223	. . 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 6685	. . . . 5 ⊢ ((x ∈ R ∧ y ∈ R ∧ ¬ (x <R y ∨ y <R x)) → x = y)
16	15	3expia 1105	. . . 4 ⊢ ((x ∈ R ∧ y ∈ R) → (¬ (x <R y ∨ y <R x) → x = y))
17		ltresr 6716	. . . . . 6 ⊢ (⟨x, 0R⟩ <ℝ ⟨y, 0R⟩ ↔ x <R y)
18		ltresr 6716	. . . . . 6 ⊢ (⟨y, 0R⟩ <ℝ ⟨x, 0R⟩ ↔ y <R x)
19	17, 18	orbi12i 680	. . . . 5 ⊢ ((⟨x, 0R⟩ <ℝ ⟨y, 0R⟩ ∨ ⟨y, 0R⟩ <ℝ ⟨x, 0R⟩) ↔ (x <R y ∨ y <R x))
20	19	notbii 593	. . . 4 ⊢ (¬ (⟨x, 0R⟩ <ℝ ⟨y, 0R⟩ ∨ ⟨y, 0R⟩ <ℝ ⟨x, 0R⟩) ↔ ¬ (x <R y ∨ y <R x))
21		vex 2554	. . . . 5 ⊢ x ∈ V
22	21	eqresr 6713	. . . 4 ⊢ (⟨x, 0R⟩ = ⟨y, 0R⟩ ↔ x = y)
23	16, 20, 22	3imtr4g 194	. . 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 2581	. 2 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (¬ (A <ℝ B ∨ B <ℝ A) → A = B))
25	24	3impia 1100	1 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ ¬ (A <ℝ B ∨ B <ℝ A)) → A = B)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∨ wo 628   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ⟨cop 3370   class class class wbr 3755  Rcnr 6281  0_Rc0r 6282   <_R cltr 6287  ℝcr 6690   <_ℝ cltrr 6695

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-i1p 6449  df-iplp 6450  df-iltp 6452  df-enr 6634  df-nr 6635  df-ltr 6638  df-0r 6639  df-r 6701  df-lt 6704

This theorem is referenced by: (None)