Theorem aptipr 7603

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 992	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → 𝐴 ∈ P)
2		simp2 993	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → 𝐵 ∈ P)
3		ioran 747	. . . . . . 7 ⊢ (¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴) ↔ (¬ 𝐴<P 𝐵 ∧ ¬ 𝐵<P 𝐴))
4	3	biimpi 119	. . . . . 6 ⊢ (¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴) → (¬ 𝐴<P 𝐵 ∧ ¬ 𝐵<P 𝐴))
5	4	3ad2ant3 1015	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (¬ 𝐴<P 𝐵 ∧ ¬ 𝐵<P 𝐴))
6	5	simprd 113	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → ¬ 𝐵<P 𝐴)
7		aptiprleml 7601	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (1st ‘𝐴) ⊆ (1st ‘𝐵))
8	1, 2, 6, 7	syl3anc 1233	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (1st ‘𝐴) ⊆ (1st ‘𝐵))
9	5	simpld 111	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → ¬ 𝐴<P 𝐵)
10		aptiprleml 7601	. . . 4 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P ∧ ¬ 𝐴<P 𝐵) → (1st ‘𝐵) ⊆ (1st ‘𝐴))
11	2, 1, 9, 10	syl3anc 1233	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (1st ‘𝐵) ⊆ (1st ‘𝐴))
12	8, 11	eqssd 3164	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (1st ‘𝐴) = (1st ‘𝐵))
13		aptiprlemu 7602	. . . 4 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P ∧ ¬ 𝐴<P 𝐵) → (2nd ‘𝐴) ⊆ (2nd ‘𝐵))
14	2, 1, 9, 13	syl3anc 1233	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (2nd ‘𝐴) ⊆ (2nd ‘𝐵))
15		aptiprlemu 7602	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (2nd ‘𝐵) ⊆ (2nd ‘𝐴))
16	1, 2, 6, 15	syl3anc 1233	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (2nd ‘𝐵) ⊆ (2nd ‘𝐴))
17	14, 16	eqssd 3164	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (2nd ‘𝐴) = (2nd ‘𝐵))
18		preqlu 7434	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 = 𝐵 ↔ ((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵))))
19	18	3adant3 1012	. 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 939	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 703   ∧ w3a 973   = wceq 1348   ∈ wcel 2141   ⊆ wss 3121   class class class wbr 3989  ‘cfv 5198  1^st c1st 6117  2^nd c2nd 6118  Pcnp 7253  <_P cltp 7257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iltp 7432

This theorem is referenced by:  aptisr  7741