Theorem preqlu 7539

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 7538	. . . . 5 |- P. C_ ( ~P Q. X. ~P Q. )
2	1	sseli 3179	. . . 4 |- ( A e. P. -> A e. ( ~P Q. X. ~P Q. ) )
3		1st2nd2 6233	. . . 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 3179	. . . 4 |- ( B e. P. -> B e. ( ~P Q. X. ~P Q. ) )
6		1st2nd2 6233	. . . 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 2212	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( A = B <-> <. ( 1st ` A ) , ( 2nd ` A ) >. = <. ( 1st ` B ) , ( 2nd ` B ) >. ) )
9		xp1st 6223	. . . . 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 6224	. . . . 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 4271	. . . 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 1364   e. wcel 2167   ~Pcpw 3605   <.cop 3625   X. cxp 4661   ` cfv 5258   1stc1st 6196   2ndc2nd 6197   Q.cnq 7347   P.cnp 7358

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fv 5266  df-1st 6198  df-2nd 6199  df-inp 7533

This theorem is referenced by:  genpassg  7593  addnqpr  7628  mulnqpr  7644  distrprg  7655  1idpr  7659  ltexpri  7680  addcanprg  7683  recexprlemex  7704  aptipr  7708