Theorem preqlu 7735

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 7734	. . . . 5 |- P. C_ ( ~P Q. X. ~P Q. )
2	1	sseli 3224	. . . 4 |- ( A e. P. -> A e. ( ~P Q. X. ~P Q. ) )
3		1st2nd2 6347	. . . 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 3224	. . . 4 |- ( B e. P. -> B e. ( ~P Q. X. ~P Q. ) )
6		1st2nd2 6347	. . . 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 2247	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( A = B <-> <. ( 1st ` A ) , ( 2nd ` A ) >. = <. ( 1st ` B ) , ( 2nd ` B ) >. ) )
9		xp1st 6337	. . . . 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 6338	. . . . 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 4336	. . . 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 1398   e. wcel 2202   ~Pcpw 3656   <.cop 3676   X. cxp 4729   ` cfv 5333   1stc1st 6310   2ndc2nd 6311   Q.cnq 7543   P.cnp 7554

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fv 5341  df-1st 6312  df-2nd 6313  df-inp 7729

This theorem is referenced by:  genpassg  7789  addnqpr  7824  mulnqpr  7840  distrprg  7851  1idpr  7855  ltexpri  7876  addcanprg  7879  recexprlemex  7900  aptipr  7904