Theorem preqlu 7691

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 7690	. . . . 5 |- P. C_ ( ~P Q. X. ~P Q. )
2	1	sseli 3223	. . . 4 |- ( A e. P. -> A e. ( ~P Q. X. ~P Q. ) )
3		1st2nd2 6337	. . . 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 3223	. . . 4 |- ( B e. P. -> B e. ( ~P Q. X. ~P Q. ) )
6		1st2nd2 6337	. . . 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 6327	. . . . 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 6328	. . . . 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 4330	. . . 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 1397   e. wcel 2202   ~Pcpw 3652   <.cop 3672   X. cxp 4723   ` cfv 5326   1stc1st 6300   2ndc2nd 6301   Q.cnq 7499   P.cnp 7510

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fv 5334  df-1st 6302  df-2nd 6303  df-inp 7685

This theorem is referenced by:  genpassg  7745  addnqpr  7780  mulnqpr  7796  distrprg  7807  1idpr  7811  ltexpri  7832  addcanprg  7835  recexprlemex  7856  aptipr  7860