Theorem preqlu 6801

Description: Two reals are equal if and only if their lower and upper cuts are. (Contributed by Jim Kingdon, 11-Dec-2019.)

Assertion

Ref	Expression
preqlu	|- ( ( A e. P. /\ B e. P. ) -> ( A = B <-> ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) ) )

Proof of Theorem preqlu

Step	Hyp	Ref	Expression
1		npsspw 6800	. . . . 5 |- P. C_ ( ~P Q. X. ~P Q. )
2	1	sseli 3005	. . . 4 |- ( A e. P. -> A e. ( ~P Q. X. ~P Q. ) )
3		1st2nd2 5854	. . . 4 |- ( A e. ( ~P Q. X. ~P Q. ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
4	2, 3	syl 14	. . 3 |- ( A e. P. -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
5	1	sseli 3005	. . . 4 |- ( B e. P. -> B e. ( ~P Q. X. ~P Q. ) )
6		1st2nd2 5854	. . . 4 |- ( B e. ( ~P Q. X. ~P Q. ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
7	5, 6	syl 14	. . 3 |- ( B e. P. -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
8	4, 7	eqeqan12d 2098	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( A = B <-> <. ( 1st ` A ) , ( 2nd ` A ) >. = <. ( 1st ` B ) , ( 2nd ` B ) >. ) )
9		xp1st 5845	. . . . 5 |- ( A e. ( ~P Q. X. ~P Q. ) -> ( 1st ` A ) e. ~P Q. )
10	2, 9	syl 14	. . . 4 |- ( A e. P. -> ( 1st ` A ) e. ~P Q. )
11		xp2nd 5846	. . . . 5 |- ( A e. ( ~P Q. X. ~P Q. ) -> ( 2nd ` A ) e. ~P Q. )
12	2, 11	syl 14	. . . 4 |- ( A e. P. -> ( 2nd ` A ) e. ~P Q. )
13		opthg 4022	. . . 4 |- ( ( ( 1st ` A ) e. ~P Q. /\ ( 2nd ` A ) e. ~P Q. ) -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. = <. ( 1st ` B ) , ( 2nd ` B ) >. <-> ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) ) )
14	10, 12, 13	syl2anc 403	. . 3 |- ( A e. P. -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. = <. ( 1st ` B ) , ( 2nd ` B ) >. <-> ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) ) )
15	14	adantr 270	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. = <. ( 1st ` B ) , ( 2nd ` B ) >. <-> ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) ) )
16	8, 15	bitrd 186	1 |- ( ( A e. P. /\ B e. P. ) -> ( A = B <-> ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   e. wcel 1434   ~Pcpw 3401   <.cop 3420   X. cxp 4390   ` cfv 4953   1stc1st 5818   2ndc2nd 5819   Q.cnq 6609   P.cnp 6620

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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2613  df-sbc 2826  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-opab 3861  df-mpt 3862  df-id 4077  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-iota 4918  df-fun 4955  df-fv 4961  df-1st 5820  df-2nd 5821  df-inp 6795

This theorem is referenced by:  genpassg  6855  addnqpr  6890  mulnqpr  6906  distrprg  6917  1idpr  6921  ltexpri  6942  addcanprg  6945  recexprlemex  6966  aptipr  6970