Theorem aptipr 7824

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 1021	. . . 4 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> A e. P. )
2		simp2 1022	. . . 4 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> B e. P. )
3		ioran 757	. . . . . . 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 1044	. . . . 5 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> ( -. A <P B /\ -. B <P A ) )
6	5	simprd 114	. . . 4 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> -. B <P A )
7		aptiprleml 7822	. . . 4 |- ( ( A e. P. /\ B e. P. /\ -. B <P A ) -> ( 1st ` A ) C_ ( 1st ` B ) )
8	1, 2, 6, 7	syl3anc 1271	. . 3 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> ( 1st ` A ) C_ ( 1st ` B ) )
9	5	simpld 112	. . . 4 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> -. A <P B )
10		aptiprleml 7822	. . . 4 |- ( ( B e. P. /\ A e. P. /\ -. A <P B ) -> ( 1st ` B ) C_ ( 1st ` A ) )
11	2, 1, 9, 10	syl3anc 1271	. . 3 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> ( 1st ` B ) C_ ( 1st ` A ) )
12	8, 11	eqssd 3241	. 2 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> ( 1st ` A ) = ( 1st ` B ) )
13		aptiprlemu 7823	. . . 4 |- ( ( B e. P. /\ A e. P. /\ -. A <P B ) -> ( 2nd ` A ) C_ ( 2nd ` B ) )
14	2, 1, 9, 13	syl3anc 1271	. . 3 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> ( 2nd ` A ) C_ ( 2nd ` B ) )
15		aptiprlemu 7823	. . . 4 |- ( ( A e. P. /\ B e. P. /\ -. B <P A ) -> ( 2nd ` B ) C_ ( 2nd ` A ) )
16	1, 2, 6, 15	syl3anc 1271	. . 3 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> ( 2nd ` B ) C_ ( 2nd ` A ) )
17	14, 16	eqssd 3241	. 2 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> ( 2nd ` A ) = ( 2nd ` B ) )
18		preqlu 7655	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( A = B <-> ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) ) )
19	18	3adant3 1041	. 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 950	1 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> A = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   C_ wss 3197   class class class wbr 4082   ` cfv 5317   1stc1st 6282   2ndc2nd 6283   P.cnp 7474   <P cltp 7478

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-eprel 4379  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-1o 6560  df-2o 6561  df-oadd 6564  df-omul 6565  df-er 6678  df-ec 6680  df-qs 6684  df-ni 7487  df-pli 7488  df-mi 7489  df-lti 7490  df-plpq 7527  df-mpq 7528  df-enq 7530  df-nqqs 7531  df-plqqs 7532  df-mqqs 7533  df-1nqqs 7534  df-rq 7535  df-ltnqqs 7536  df-enq0 7607  df-nq0 7608  df-0nq0 7609  df-plq0 7610  df-mq0 7611  df-inp 7649  df-iltp 7653

This theorem is referenced by:  aptisr  7962