Theorem aptisr 7611

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 7559	. . 3 ⊢ R = ((P × P) / ~R )
2		breq1 3940	. . . . . 6 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝐴 <R [⟨𝑧, 𝑤⟩] ~R ))
3		breq2 3941	. . . . . 6 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ [⟨𝑧, 𝑤⟩] ~R <R 𝐴))
4	2, 3	orbi12d 783	. . . . 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 657	. . . 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 2147	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ↔ 𝐴 = [⟨𝑧, 𝑤⟩] ~R ))
7	5, 6	imbi12d 233	. . 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 3941	. . . . . 6 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝐴 <R 𝐵))
9		breq1 3940	. . . . . 6 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ([⟨𝑧, 𝑤⟩] ~R <R 𝐴 ↔ 𝐵 <R 𝐴))
10	8, 9	orbi12d 783	. . . . 5 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((𝐴 <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝐴) ↔ (𝐴 <R 𝐵 ∨ 𝐵 <R 𝐴)))
11	10	notbid 657	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (¬ (𝐴 <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝐴) ↔ ¬ (𝐴 <R 𝐵 ∨ 𝐵 <R 𝐴)))
12		eqeq2 2150	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 = [⟨𝑧, 𝑤⟩] ~R ↔ 𝐴 = 𝐵))
13	11, 12	imbi12d 233	. . 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 7410	. . . . . . . . 9 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦 +P 𝑧) = (𝑧 +P 𝑦))
15	14	ad2ant2lr 502	. . . . . . . 8 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑦 +P 𝑧) = (𝑧 +P 𝑦))
16		addcomprg 7410	. . . . . . . . 9 ⊢ ((𝑥 ∈ P ∧ 𝑤 ∈ P) → (𝑥 +P 𝑤) = (𝑤 +P 𝑥))
17	16	ad2ant2rl 503	. . . . . . . 8 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑥 +P 𝑤) = (𝑤 +P 𝑥))
18	15, 17	breq12d 3950	. . . . . . 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 780	. . . . . 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 657	. . . . 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 7369	. . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑤 ∈ P) → (𝑥 +P 𝑤) ∈ P)
22	21	ad2ant2rl 503	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑥 +P 𝑤) ∈ P)
23		addclpr 7369	. . . . . . 7 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦 +P 𝑧) ∈ P)
24	23	ad2ant2lr 502	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑦 +P 𝑧) ∈ P)
25		aptipr 7473	. . . . . . 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 1184	. . . . . 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 409	. . . . 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 169	. . . 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 7579	. . . . . 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 7579	. . . . . . 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 266	. . . . . 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 783	. . . . 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 657	. . . 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 7568	. . . 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 202	. . 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 6525	. 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (¬ (𝐴 <R 𝐵 ∨ 𝐵 <R 𝐴) → 𝐴 = 𝐵))
37	36	3impia 1179	1 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ ¬ (𝐴 <R 𝐵 ∨ 𝐵 <R 𝐴)) → 𝐴 = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  ⟨cop 3535   class class class wbr 3937  (class class class)co 5782  [cec 6435  Pcnp 7123   +_P cpp 7125  <_P cltp 7127   ~_R cer 7128  Rcnr 7129   <_R cltr 7135

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-2o 6322  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-enq0 7256  df-nq0 7257  df-0nq0 7258  df-plq0 7259  df-mq0 7260  df-inp 7298  df-iplp 7300  df-iltp 7302  df-enr 7558  df-nr 7559  df-ltr 7562

This theorem is referenced by:  axpre-apti  7717