Theorem prop 7184

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 7180	. . . 4 ⊢ P ⊆ (𝒫 Q × 𝒫 Q)
2	1	sseli 3043	. . 3 ⊢ (𝐴 ∈ P → 𝐴 ∈ (𝒫 Q × 𝒫 Q))
3		1st2nd2 6003	. . 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 2162	. . 3 ⊢ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ → (𝐴 ∈ P ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P))
6	5	biimpcd 158	. 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)

Syntax hints:   → wi 4   = wceq 1299   ∈ wcel 1448  𝒫 cpw 3457  ⟨cop 3477   × cxp 4475  ‘cfv 5059  1^st c1st 5967  2^nd c2nd 5968  Qcnq 6989  Pcnp 7000

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-iota 5024  df-fun 5061  df-fv 5067  df-1st 5969  df-2nd 5970  df-inp 7175

This theorem is referenced by:  elnp1st2nd  7185  0npr  7192  genpdf  7217  genipv  7218  genpelvl  7221  genpelvu  7222  genpml  7226  genpmu  7227  genprndl  7230  genprndu  7231  genpdisj  7232  genpassl  7233  genpassu  7234  addnqprl  7238  addnqpru  7239  addlocprlemeqgt  7241  addlocprlemgt  7243  addlocprlem  7244  addlocpr  7245  nqprl  7260  nqpru  7261  addnqprlemfl  7268  addnqprlemfu  7269  mulnqprl  7277  mulnqpru  7278  mullocprlem  7279  mullocpr  7280  mulnqprlemfl  7284  mulnqprlemfu  7285  addcomprg  7287  mulcomprg  7289  distrlem1prl  7291  distrlem1pru  7292  distrlem4prl  7293  distrlem4pru  7294  ltprordil  7298  1idprl  7299  1idpru  7300  ltpopr  7304  ltsopr  7305  ltaddpr  7306  ltexprlemm  7309  ltexprlemopl  7310  ltexprlemlol  7311  ltexprlemopu  7312  ltexprlemupu  7313  ltexprlemdisj  7315  ltexprlemloc  7316  ltexprlemfl  7318  ltexprlemrl  7319  ltexprlemfu  7320  ltexprlemru  7321  addcanprleml  7323  addcanprlemu  7324  prplnqu  7329  recexprlemm  7333  recexprlemdisj  7339  recexprlemloc  7340  recexprlem1ssl  7342  recexprlem1ssu  7343  recexprlemss1l  7344  recexprlemss1u  7345  aptiprleml  7348  aptiprlemu  7349  archpr  7352  cauappcvgprlemladdru  7365  cauappcvgprlemladdrl  7366  archrecpr  7373  caucvgprlemladdrl  7387  caucvgprprlemml  7403  caucvgprprlemmu  7404  caucvgprprlemopl  7406