Theorem aptipr 7854

Description: Apartness of positive reals is tight. (Contributed by Jim Kingdon, 28-Jan-2020.)

Assertion

Ref	Expression
aptipr	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → 𝐴 = 𝐵)

Proof of Theorem aptipr

Step	Hyp	Ref	Expression
1		simp1 1021	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → 𝐴 ∈ P)
2		simp2 1022	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → 𝐵 ∈ P)
3		ioran 757	. . . . . . 7 ⊢ (¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴) ↔ (¬ 𝐴<P 𝐵 ∧ ¬ 𝐵<P 𝐴))
4	3	biimpi 120	. . . . . 6 ⊢ (¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴) → (¬ 𝐴<P 𝐵 ∧ ¬ 𝐵<P 𝐴))
5	4	3ad2ant3 1044	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (¬ 𝐴<P 𝐵 ∧ ¬ 𝐵<P 𝐴))
6	5	simprd 114	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → ¬ 𝐵<P 𝐴)
7		aptiprleml 7852	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (1st ‘𝐴) ⊆ (1st ‘𝐵))
8	1, 2, 6, 7	syl3anc 1271	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (1st ‘𝐴) ⊆ (1st ‘𝐵))
9	5	simpld 112	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → ¬ 𝐴<P 𝐵)
10		aptiprleml 7852	. . . 4 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P ∧ ¬ 𝐴<P 𝐵) → (1st ‘𝐵) ⊆ (1st ‘𝐴))
11	2, 1, 9, 10	syl3anc 1271	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (1st ‘𝐵) ⊆ (1st ‘𝐴))
12	8, 11	eqssd 3242	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (1st ‘𝐴) = (1st ‘𝐵))
13		aptiprlemu 7853	. . . 4 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P ∧ ¬ 𝐴<P 𝐵) → (2nd ‘𝐴) ⊆ (2nd ‘𝐵))
14	2, 1, 9, 13	syl3anc 1271	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (2nd ‘𝐴) ⊆ (2nd ‘𝐵))
15		aptiprlemu 7853	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (2nd ‘𝐵) ⊆ (2nd ‘𝐴))
16	1, 2, 6, 15	syl3anc 1271	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (2nd ‘𝐵) ⊆ (2nd ‘𝐴))
17	14, 16	eqssd 3242	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (2nd ‘𝐴) = (2nd ‘𝐵))
18		preqlu 7685	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 = 𝐵 ↔ ((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵))))
19	18	3adant3 1041	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (𝐴 = 𝐵 ↔ ((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵))))
20	12, 17, 19	mpbir2and 950	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200   ⊆ wss 3198   class class class wbr 4086  ‘cfv 5324  1^st c1st 6296  2^nd c2nd 6297  Pcnp 7504  <_P cltp 7508

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 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-2o 6578  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7517  df-pli 7518  df-mi 7519  df-lti 7520  df-plpq 7557  df-mpq 7558  df-enq 7560  df-nqqs 7561  df-plqqs 7562  df-mqqs 7563  df-1nqqs 7564  df-rq 7565  df-ltnqqs 7566  df-enq0 7637  df-nq0 7638  df-0nq0 7639  df-plq0 7640  df-mq0 7641  df-inp 7679  df-iltp 7683

This theorem is referenced by:  aptisr  7992