Theorem aptipr 7708

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 999	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → 𝐴 ∈ P)
2		simp2 1000	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → 𝐵 ∈ P)
3		ioran 753	. . . . . . 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 1022	. . . . 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 7706	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (1st ‘𝐴) ⊆ (1st ‘𝐵))
8	1, 2, 6, 7	syl3anc 1249	. . 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 7706	. . . 4 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P ∧ ¬ 𝐴<P 𝐵) → (1st ‘𝐵) ⊆ (1st ‘𝐴))
11	2, 1, 9, 10	syl3anc 1249	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (1st ‘𝐵) ⊆ (1st ‘𝐴))
12	8, 11	eqssd 3200	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (1st ‘𝐴) = (1st ‘𝐵))
13		aptiprlemu 7707	. . . 4 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P ∧ ¬ 𝐴<P 𝐵) → (2nd ‘𝐴) ⊆ (2nd ‘𝐵))
14	2, 1, 9, 13	syl3anc 1249	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (2nd ‘𝐴) ⊆ (2nd ‘𝐵))
15		aptiprlemu 7707	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (2nd ‘𝐵) ⊆ (2nd ‘𝐴))
16	1, 2, 6, 15	syl3anc 1249	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (2nd ‘𝐵) ⊆ (2nd ‘𝐴))
17	14, 16	eqssd 3200	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (2nd ‘𝐴) = (2nd ‘𝐵))
18		preqlu 7539	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 = 𝐵 ↔ ((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵))))
19	18	3adant3 1019	. 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 946	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   ∧ w3a 980   = wceq 1364   ∈ wcel 2167   ⊆ wss 3157   class class class wbr 4033  ‘cfv 5258  1^st c1st 6196  2^nd c2nd 6197  Pcnp 7358  <_P cltp 7362

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-2o 6475  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420  df-enq0 7491  df-nq0 7492  df-0nq0 7493  df-plq0 7494  df-mq0 7495  df-inp 7533  df-iltp 7537

This theorem is referenced by:  aptisr  7846