Theorem preqlu 7655

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 7654	. . . . 5 |- P. C_ ( ~P Q. X. ~P Q. )
2	1	sseli 3220	. . . 4 |- ( A e. P. -> A e. ( ~P Q. X. ~P Q. ) )
3		1st2nd2 6319	. . . 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 3220	. . . 4 |- ( B e. P. -> B e. ( ~P Q. X. ~P Q. ) )
6		1st2nd2 6319	. . . 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 2245	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( A = B <-> <. ( 1st ` A ) , ( 2nd ` A ) >. = <. ( 1st ` B ) , ( 2nd ` B ) >. ) )
9		xp1st 6309	. . . . 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 6310	. . . . 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 4323	. . . 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 411	. . 3 |- ( A e. P. -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. = <. ( 1st ` B ) , ( 2nd ` B ) >. <-> ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) ) )
15	14	adantr 276	. 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 188	1 |- ( ( A e. P. /\ B e. P. ) -> ( A = B <-> ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   ~Pcpw 3649   <.cop 3669   X. cxp 4716   ` cfv 5317   1stc1st 6282   2ndc2nd 6283   Q.cnq 7463   P.cnp 7474

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-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-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  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-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-iota 5277  df-fun 5319  df-fv 5325  df-1st 6284  df-2nd 6285  df-inp 7649

This theorem is referenced by:  genpassg  7709  addnqpr  7744  mulnqpr  7760  distrprg  7771  1idpr  7775  ltexpri  7796  addcanprg  7799  recexprlemex  7820  aptipr  7824