Theorem prop 7843

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 7839	. . . 4 ⊢ P ⊆ (𝒫 Q × 𝒫 Q)
2	1	sseli 3244	. . 3 ⊢ (𝐴 ∈ P → 𝐴 ∈ (𝒫 Q × 𝒫 Q))
3		1st2nd2 6409	. . 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 4772  ‘cfv 5377  1^st c1st 6372  2^nd c2nd 6373  Qcnq 7648  Pcnp 7659

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 7834

This theorem is used by:  elnp1st2nd  7844  0npr  7851  genpdf  7876  genipv  7877  genpelvl  7880  genpelvu  7881  genpml  7885  genpmu  7886  genprndl  7889  genprndu  7890  genpdisj  7891  genpassl  7892  genpassu  7893  addnqprl  7897  addnqpru  7898  addlocprlemeqgt  7900  addlocprlemgt  7902  addlocprlem  7903  addlocpr  7904  nqprl  7919  nqpru  7920  addnqprlemfl  7927  addnqprlemfu  7928  mulnqprl  7936  mulnqpru  7937  mullocprlem  7938  mullocpr  7939  mulnqprlemfl  7943  mulnqprlemfu  7944  addcomprg  7946  mulcomprg  7948  distrlem1prl  7950  distrlem1pru  7951  distrlem4prl  7952  distrlem4pru  7953  ltprordil  7957  1idprl  7958  1idpru  7959  ltpopr  7963  ltsopr  7964  ltaddpr  7965  ltexprlemm  7968  ltexprlemopl  7969  ltexprlemlol  7970  ltexprlemopu  7971  ltexprlemupu  7972  ltexprlemdisj  7974  ltexprlemloc  7975  ltexprlemfl  7977  ltexprlemrl  7978  ltexprlemfu  7979  ltexprlemru  7980  addcanprleml  7982  addcanprlemu  7983  prplnqu  7988  recexprlemm  7992  recexprlemdisj  7998  recexprlemloc  7999  recexprlem1ssl  8001  recexprlem1ssu  8002  recexprlemss1l  8003  recexprlemss1u  8004  aptiprleml  8007  aptiprlemu  8008  archpr  8011  cauappcvgprlemladdru  8024  cauappcvgprlemladdrl  8025  archrecpr  8032  caucvgprlemladdrl  8046  caucvgprprlemml  8062  caucvgprprlemmu  8063  caucvgprprlemopl  8065  suplocexprlemml  8084  suplocexprlemrl  8085  suplocexprlemmu  8086  suplocexprlemdisj  8088  suplocexprlemloc  8089  suplocexprlemex  8090  suplocexprlemub  8091