Theorem aptisr 6955

Description: Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.)

Assertion

Ref	Expression
aptisr	⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ ¬ (𝐴 <R 𝐵 ∨ 𝐵 <R 𝐴)) → 𝐴 = 𝐵)

Proof of Theorem aptisr

Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nr 6904	. . 3 ⊢ R = ((P × P) / ~R )
2		breq1 3788	. . . . . 6 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝐴 <R [⟨𝑧, 𝑤⟩] ~R ))
3		breq2 3789	. . . . . 6 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ [⟨𝑧, 𝑤⟩] ~R <R 𝐴))
4	2, 3	orbi12d 739	. . . . 5 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ) ↔ (𝐴 <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝐴)))
5	4	notbid 624	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (¬ ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ) ↔ ¬ (𝐴 <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝐴)))
6		eqeq1 2087	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ↔ 𝐴 = [⟨𝑧, 𝑤⟩] ~R ))
7	5, 6	imbi12d 232	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ((¬ ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ) → [⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ) ↔ (¬ (𝐴 <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝐴) → 𝐴 = [⟨𝑧, 𝑤⟩] ~R )))
8		breq2 3789	. . . . . 6 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝐴 <R 𝐵))
9		breq1 3788	. . . . . 6 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ([⟨𝑧, 𝑤⟩] ~R <R 𝐴 ↔ 𝐵 <R 𝐴))
10	8, 9	orbi12d 739	. . . . 5 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((𝐴 <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝐴) ↔ (𝐴 <R 𝐵 ∨ 𝐵 <R 𝐴)))
11	10	notbid 624	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (¬ (𝐴 <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝐴) ↔ ¬ (𝐴 <R 𝐵 ∨ 𝐵 <R 𝐴)))
12		eqeq2 2090	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 = [⟨𝑧, 𝑤⟩] ~R ↔ 𝐴 = 𝐵))
13	11, 12	imbi12d 232	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((¬ (𝐴 <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝐴) → 𝐴 = [⟨𝑧, 𝑤⟩] ~R ) ↔ (¬ (𝐴 <R 𝐵 ∨ 𝐵 <R 𝐴) → 𝐴 = 𝐵)))
14		addcomprg 6768	. . . . . . . . 9 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦 +P 𝑧) = (𝑧 +P 𝑦))
15	14	ad2ant2lr 493	. . . . . . . 8 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑦 +P 𝑧) = (𝑧 +P 𝑦))
16		addcomprg 6768	. . . . . . . . 9 ⊢ ((𝑥 ∈ P ∧ 𝑤 ∈ P) → (𝑥 +P 𝑤) = (𝑤 +P 𝑥))
17	16	ad2ant2rl 494	. . . . . . . 8 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑥 +P 𝑤) = (𝑤 +P 𝑥))
18	15, 17	breq12d 3798	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑦 +P 𝑧)<P (𝑥 +P 𝑤) ↔ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥)))
19	18	orbi2d 736	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ∨ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤)) ↔ ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ∨ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥))))
20	19	notbid 624	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (¬ ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ∨ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤)) ↔ ¬ ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ∨ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥))))
21		addclpr 6727	. . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑤 ∈ P) → (𝑥 +P 𝑤) ∈ P)
22	21	ad2ant2rl 494	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑥 +P 𝑤) ∈ P)
23		addclpr 6727	. . . . . . 7 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦 +P 𝑧) ∈ P)
24	23	ad2ant2lr 493	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑦 +P 𝑧) ∈ P)
25		aptipr 6831	. . . . . . 7 ⊢ (((𝑥 +P 𝑤) ∈ P ∧ (𝑦 +P 𝑧) ∈ P ∧ ¬ ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ∨ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤))) → (𝑥 +P 𝑤) = (𝑦 +P 𝑧))
26	25	3expia 1140	. . . . . 6 ⊢ (((𝑥 +P 𝑤) ∈ P ∧ (𝑦 +P 𝑧) ∈ P) → (¬ ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ∨ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤)) → (𝑥 +P 𝑤) = (𝑦 +P 𝑧)))
27	22, 24, 26	syl2anc 403	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (¬ ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ∨ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤)) → (𝑥 +P 𝑤) = (𝑦 +P 𝑧)))
28	20, 27	sylbird 168	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (¬ ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ∨ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥)) → (𝑥 +P 𝑤) = (𝑦 +P 𝑧)))
29		ltsrprg 6924	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧)))
30		ltsrprg 6924	. . . . . . 7 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑥 ∈ P ∧ 𝑦 ∈ P)) → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥)))
31	30	ancoms 264	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥)))
32	29, 31	orbi12d 739	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ) ↔ ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ∨ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥))))
33	32	notbid 624	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (¬ ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ) ↔ ¬ ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ∨ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥))))
34		enreceq 6913	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤) = (𝑦 +P 𝑧)))
35	28, 33, 34	3imtr4d 201	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (¬ ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ) → [⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ))
36	1, 7, 13, 35	2ecoptocl 6217	. 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (¬ (𝐴 <R 𝐵 ∨ 𝐵 <R 𝐴) → 𝐴 = 𝐵))
37	36	3impia 1135	1 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ ¬ (𝐴 <R 𝐵 ∨ 𝐵 <R 𝐴)) → 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 661   ∧ w3a 919   = wceq 1284   ∈ wcel 1433  ⟨cop 3401   class class class wbr 3785  (class class class)co 5532  [cec 6127  Pcnp 6481   +_P cpp 6483  <_P cltp 6485   ~_R cer 6486  Rcnr 6487   <_R cltr 6493

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-2o 6025  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-enq0 6614  df-nq0 6615  df-0nq0 6616  df-plq0 6617  df-mq0 6618  df-inp 6656  df-iplp 6658  df-iltp 6660  df-enr 6903  df-nr 6904  df-ltr 6907

This theorem is referenced by:  axpre-apti  7051