Theorem aptipr 7582

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 987	. . . 4 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> A e. P. )
2		simp2 988	. . . 4 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> B e. P. )
3		ioran 742	. . . . . . 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 1010	. . . . 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 7580	. . . 4 |- ( ( A e. P. /\ B e. P. /\ -. B <P A ) -> ( 1st ` A ) C_ ( 1st ` B ) )
8	1, 2, 6, 7	syl3anc 1228	. . 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 7580	. . . 4 |- ( ( B e. P. /\ A e. P. /\ -. A <P B ) -> ( 1st ` B ) C_ ( 1st ` A ) )
11	2, 1, 9, 10	syl3anc 1228	. . 3 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> ( 1st ` B ) C_ ( 1st ` A ) )
12	8, 11	eqssd 3159	. 2 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> ( 1st ` A ) = ( 1st ` B ) )
13		aptiprlemu 7581	. . . 4 |- ( ( B e. P. /\ A e. P. /\ -. A <P B ) -> ( 2nd ` A ) C_ ( 2nd ` B ) )
14	2, 1, 9, 13	syl3anc 1228	. . 3 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> ( 2nd ` A ) C_ ( 2nd ` B ) )
15		aptiprlemu 7581	. . . 4 |- ( ( A e. P. /\ B e. P. /\ -. B <P A ) -> ( 2nd ` B ) C_ ( 2nd ` A ) )
16	1, 2, 6, 15	syl3anc 1228	. . 3 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> ( 2nd ` B ) C_ ( 2nd ` A ) )
17	14, 16	eqssd 3159	. 2 |- ( ( A e. P. /\ B e. P. /\ -. ( A <P B \/ B <P A ) ) -> ( 2nd ` A ) = ( 2nd ` B ) )
18		preqlu 7413	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( A = B <-> ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) ) )
19	18	3adant3 1007	. 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 934	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 698   /\ w3a 968   = wceq 1343   e. wcel 2136   C_ wss 3116   class class class wbr 3982   ` cfv 5188   1stc1st 6106   2ndc2nd 6107   P.cnp 7232   <P cltp 7236

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-iltp 7411

This theorem is referenced by:  aptisr  7720