Theorem prop 7832

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 7828	. . . 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 6399	. . 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. )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   ~Pcpw 3685   <.cop 3708   X. cxp 4767   ` cfv 5372   1stc1st 6362   2ndc2nd 6363   Q.cnq 7637   P.cnp 7648

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 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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fv 5380  df-1st 6364  df-2nd 6365  df-inp 7823

This theorem is referenced by:  elnp1st2nd  7833  0npr  7840  genpdf  7865  genipv  7866  genpelvl  7869  genpelvu  7870  genpml  7874  genpmu  7875  genprndl  7878  genprndu  7879  genpdisj  7880  genpassl  7881  genpassu  7882  addnqprl  7886  addnqpru  7887  addlocprlemeqgt  7889  addlocprlemgt  7891  addlocprlem  7892  addlocpr  7893  nqprl  7908  nqpru  7909  addnqprlemfl  7916  addnqprlemfu  7917  mulnqprl  7925  mulnqpru  7926  mullocprlem  7927  mullocpr  7928  mulnqprlemfl  7932  mulnqprlemfu  7933  addcomprg  7935  mulcomprg  7937  distrlem1prl  7939  distrlem1pru  7940  distrlem4prl  7941  distrlem4pru  7942  ltprordil  7946  1idprl  7947  1idpru  7948  ltpopr  7952  ltsopr  7953  ltaddpr  7954  ltexprlemm  7957  ltexprlemopl  7958  ltexprlemlol  7959  ltexprlemopu  7960  ltexprlemupu  7961  ltexprlemdisj  7963  ltexprlemloc  7964  ltexprlemfl  7966  ltexprlemrl  7967  ltexprlemfu  7968  ltexprlemru  7969  addcanprleml  7971  addcanprlemu  7972  prplnqu  7977  recexprlemm  7981  recexprlemdisj  7987  recexprlemloc  7988  recexprlem1ssl  7990  recexprlem1ssu  7991  recexprlemss1l  7992  recexprlemss1u  7993  aptiprleml  7996  aptiprlemu  7997  archpr  8000  cauappcvgprlemladdru  8013  cauappcvgprlemladdrl  8014  archrecpr  8021  caucvgprlemladdrl  8035  caucvgprprlemml  8051  caucvgprprlemmu  8052  caucvgprprlemopl  8054  suplocexprlemml  8073  suplocexprlemrl  8074  suplocexprlemmu  8075  suplocexprlemdisj  8077  suplocexprlemloc  8078  suplocexprlemex  8079  suplocexprlemub  8080