Theorem prop 7848

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 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)

Proof of Theorem prop

Step	Hyp	Ref	Expression
1		npsspw 7844	. . . 4 ⊢ P ⊆ (𝒫 Q × 𝒫 Q)
2	1	sseli 3244	. . 3 ⊢ (𝐴 ∈ P → 𝐴 ∈ (𝒫 Q × 𝒫 Q))
3		1st2nd2 6413	. . 3 ⊢ (𝐴 ∈ (𝒫 Q × 𝒫 Q) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
4	2, 3	syl 14	. 2 ⊢ (𝐴 ∈ P → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
5		eleq1 2301	. . 3 ⊢ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ → (𝐴 ∈ P ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P))
6	5	biimpcd 159	. 2 ⊢ (𝐴 ∈ P → (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P))
7	4, 6	mpd 13	1 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∈ wcel 2209  𝒫 cpw 3688  ⟨cop 3712   × cxp 4774  ‘cfv 5380  1^st c1st 6376  2^nd c2nd 6377  Qcnq 7653  Pcnp 7664

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 4251  ax-pow 4313  ax-pr 4348  ax-un 4580

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 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-iota 5340  df-fun 5382  df-fv 5388  df-1st 6378  df-2nd 6379  df-inp 7839

This theorem is used by:  elnp1st2nd  7849  0npr  7856  genpdf  7881  genipv  7882  genpelvl  7885  genpelvu  7886  genpml  7890  genpmu  7891  genprndl  7894  genprndu  7895  genpdisj  7896  genpassl  7897  genpassu  7898  addnqprl  7902  addnqpru  7903  addlocprlemeqgt  7905  addlocprlemgt  7907  addlocprlem  7908  addlocpr  7909  nqprl  7924  nqpru  7925  addnqprlemfl  7932  addnqprlemfu  7933  mulnqprl  7941  mulnqpru  7942  mullocprlem  7943  mullocpr  7944  mulnqprlemfl  7948  mulnqprlemfu  7949  addcomprg  7951  mulcomprg  7953  distrlem1prl  7955  distrlem1pru  7956  distrlem4prl  7957  distrlem4pru  7958  ltprordil  7962  1idprl  7963  1idpru  7964  ltpopr  7968  ltsopr  7969  ltaddpr  7970  ltexprlemm  7973  ltexprlemopl  7974  ltexprlemlol  7975  ltexprlemopu  7976  ltexprlemupu  7977  ltexprlemdisj  7979  ltexprlemloc  7980  ltexprlemfl  7982  ltexprlemrl  7983  ltexprlemfu  7984  ltexprlemru  7985  addcanprleml  7987  addcanprlemu  7988  prplnqu  7993  recexprlemm  7997  recexprlemdisj  8003  recexprlemloc  8004  recexprlem1ssl  8006  recexprlem1ssu  8007  recexprlemss1l  8008  recexprlemss1u  8009  aptiprleml  8012  aptiprlemu  8013  archpr  8016  cauappcvgprlemladdru  8029  cauappcvgprlemladdrl  8030  archrecpr  8037  caucvgprlemladdrl  8051  caucvgprprlemml  8067  caucvgprprlemmu  8068  caucvgprprlemopl  8070  suplocexprlemml  8089  suplocexprlemrl  8090  suplocexprlemmu  8091  suplocexprlemdisj  8093  suplocexprlemloc  8094  suplocexprlemex  8095  suplocexprlemub  8096