Theorem aptipr 7866

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 1023	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → 𝐴 ∈ P)
2		simp2 1024	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → 𝐵 ∈ P)
3		ioran 759	. . . . . . 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 1046	. . . . 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 7864	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (1st ‘𝐴) ⊆ (1st ‘𝐵))
8	1, 2, 6, 7	syl3anc 1273	. . 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 7864	. . . 4 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P ∧ ¬ 𝐴<P 𝐵) → (1st ‘𝐵) ⊆ (1st ‘𝐴))
11	2, 1, 9, 10	syl3anc 1273	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (1st ‘𝐵) ⊆ (1st ‘𝐴))
12	8, 11	eqssd 3243	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (1st ‘𝐴) = (1st ‘𝐵))
13		aptiprlemu 7865	. . . 4 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P ∧ ¬ 𝐴<P 𝐵) → (2nd ‘𝐴) ⊆ (2nd ‘𝐵))
14	2, 1, 9, 13	syl3anc 1273	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (2nd ‘𝐴) ⊆ (2nd ‘𝐵))
15		aptiprlemu 7865	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (2nd ‘𝐵) ⊆ (2nd ‘𝐴))
16	1, 2, 6, 15	syl3anc 1273	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (2nd ‘𝐵) ⊆ (2nd ‘𝐴))
17	14, 16	eqssd 3243	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (2nd ‘𝐴) = (2nd ‘𝐵))
18		preqlu 7697	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 = 𝐵 ↔ ((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵))))
19	18	3adant3 1043	. 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 952	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 715   ∧ w3a 1004   = wceq 1397   ∈ wcel 2201   ⊆ wss 3199   class class class wbr 4089  ‘cfv 5328  1^st c1st 6306  2^nd c2nd 6307  Pcnp 7516  <_P cltp 7520

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-eprel 4388  df-id 4392  df-po 4395  df-iso 4396  df-iord 4465  df-on 4467  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-recs 6476  df-irdg 6541  df-1o 6587  df-2o 6588  df-oadd 6591  df-omul 6592  df-er 6707  df-ec 6709  df-qs 6713  df-ni 7529  df-pli 7530  df-mi 7531  df-lti 7532  df-plpq 7569  df-mpq 7570  df-enq 7572  df-nqqs 7573  df-plqqs 7574  df-mqqs 7575  df-1nqqs 7576  df-rq 7577  df-ltnqqs 7578  df-enq0 7649  df-nq0 7650  df-0nq0 7651  df-plq0 7652  df-mq0 7653  df-inp 7691  df-iltp 7695

This theorem is referenced by:  aptisr  8004