Theorem aptipr 7261

Description: Apartness of positive reals is tight. (Contributed by Jim Kingdon, 28-Jan-2020.)

Assertion

Ref	Expression
aptipr	|- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> A = B )

Proof of Theorem aptipr

Step	Hyp	Ref	Expression
1		simp1 944	. . . 4 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> A e. P. )
2		simp2 945	. . . 4 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> B e. P. )
3		ioran 705	. . . . . . 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 967	. . . . 5 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> ( -. A <P B /\ -. B <P A ) )
6	5	simprd 113	. . . 4 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> -. B <P A )
7		aptiprleml 7259	. . . 4 |- ( ( A e. P. /\ B e. P. /\ -. B <P A ) -> ( 1st ` A ) C_ ( 1st ` B ) )
8	1, 2, 6, 7	syl3anc 1175	. . 3 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> ( 1st ` A ) C_ ( 1st ` B ) )
9	5	simpld 111	. . . 4 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> -. A <P B )
10		aptiprleml 7259	. . . 4 |- ( ( B e. P. /\ A e. P. /\ -. A <P B ) -> ( 1st ` B ) C_ ( 1st ` A ) )
11	2, 1, 9, 10	syl3anc 1175	. . 3 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> ( 1st ` B ) C_ ( 1st ` A ) )
12	8, 11	eqssd 3043	. 2 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> ( 1st ` A ) = ( 1st ` B ) )
13		aptiprlemu 7260	. . . 4 |- ( ( B e. P. /\ A e. P. /\ -. A <P B ) -> ( 2nd ` A ) C_ ( 2nd ` B ) )
14	2, 1, 9, 13	syl3anc 1175	. . 3 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> ( 2nd ` A ) C_ ( 2nd ` B ) )
15		aptiprlemu 7260	. . . 4 |- ( ( A e. P. /\ B e. P. /\ -. B <P A ) -> ( 2nd ` B ) C_ ( 2nd ` A ) )
16	1, 2, 6, 15	syl3anc 1175	. . 3 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> ( 2nd ` B ) C_ ( 2nd ` A ) )
17	14, 16	eqssd 3043	. 2 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> ( 2nd ` A ) = ( 2nd ` B ) )
18		preqlu 7092	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( A = B <-> ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) ) )
19	18	3adant3 964	. 2 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> ( A = B <-> ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) ) )
20	12, 17, 19	mpbir2and 891	1 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> A = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 665   /\ w3a 925   = wceq 1290   e. wcel 1439   C_ wss 3000   class class class wbr 3851   ` cfv 5028   1stc1st 5923   2ndc2nd 5924   P.cnp 6911   <P cltp 6915

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-eprel 4125  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-1o 6195  df-2o 6196  df-oadd 6199  df-omul 6200  df-er 6306  df-ec 6308  df-qs 6312  df-ni 6924  df-pli 6925  df-mi 6926  df-lti 6927  df-plpq 6964  df-mpq 6965  df-enq 6967  df-nqqs 6968  df-plqqs 6969  df-mqqs 6970  df-1nqqs 6971  df-rq 6972  df-ltnqqs 6973  df-enq0 7044  df-nq0 7045  df-0nq0 7046  df-plq0 7047  df-mq0 7048  df-inp 7086  df-iltp 7090

This theorem is referenced by:  aptisr  7385