Theorem prop 7842

Description: A positive real is an ordered pair of a lower cut and an upper cut. (Contributed by Jim Kingdon, 27-Sep-2019.)

Assertion

Ref	Expression
prop	|- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )

Proof of Theorem prop

Step	Hyp	Ref	Expression
1		npsspw 7838	. . . 4 |- P. C_ ( ~P Q. X. ~P Q. )
2	1	sseli 3244	. . 3 |- ( A e. P. -> A e. ( ~P Q. X. ~P Q. ) )
3		1st2nd2 6409	. . 3 |- ( A e. ( ~P Q. X. ~P Q. ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
4	2, 3	syl 14	. 2 |- ( A e. P. -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
5		eleq1 2301	. . 3 |- ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. -> ( A e. P. <-> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. ) )
6	5	biimpcd 159	. 2 |- ( A e. P. -> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. ) )
7	4, 6	mpd 13	1 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   ~Pcpw 3688   <.cop 3712   X. cxp 4772   ` cfv 5377   1stc1st 6372   2ndc2nd 6373   Q.cnq 7647   P.cnp 7658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fv 5385  df-1st 6374  df-2nd 6375  df-inp 7833

This theorem is used by:  elnp1st2nd  7843  0npr  7850  genpdf  7875  genipv  7876  genpelvl  7879  genpelvu  7880  genpml  7884  genpmu  7885  genprndl  7888  genprndu  7889  genpdisj  7890  genpassl  7891  genpassu  7892  addnqprl  7896  addnqpru  7897  addlocprlemeqgt  7899  addlocprlemgt  7901  addlocprlem  7902  addlocpr  7903  nqprl  7918  nqpru  7919  addnqprlemfl  7926  addnqprlemfu  7927  mulnqprl  7935  mulnqpru  7936  mullocprlem  7937  mullocpr  7938  mulnqprlemfl  7942  mulnqprlemfu  7943  addcomprg  7945  mulcomprg  7947  distrlem1prl  7949  distrlem1pru  7950  distrlem4prl  7951  distrlem4pru  7952  ltprordil  7956  1idprl  7957  1idpru  7958  ltpopr  7962  ltsopr  7963  ltaddpr  7964  ltexprlemm  7967  ltexprlemopl  7968  ltexprlemlol  7969  ltexprlemopu  7970  ltexprlemupu  7971  ltexprlemdisj  7973  ltexprlemloc  7974  ltexprlemfl  7976  ltexprlemrl  7977  ltexprlemfu  7978  ltexprlemru  7979  addcanprleml  7981  addcanprlemu  7982  prplnqu  7987  recexprlemm  7991  recexprlemdisj  7997  recexprlemloc  7998  recexprlem1ssl  8000  recexprlem1ssu  8001  recexprlemss1l  8002  recexprlemss1u  8003  aptiprleml  8006  aptiprlemu  8007  archpr  8010  cauappcvgprlemladdru  8023  cauappcvgprlemladdrl  8024  archrecpr  8031  caucvgprlemladdrl  8045  caucvgprprlemml  8061  caucvgprprlemmu  8062  caucvgprprlemopl  8064  suplocexprlemml  8083  suplocexprlemrl  8084  suplocexprlemmu  8085  suplocexprlemdisj  8087  suplocexprlemloc  8088  suplocexprlemex  8089  suplocexprlemub  8090