Theorem prop 7836

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 7832	. . . 4 ⊢ P ⊆ (𝒫 Q × 𝒫 Q)
2	1	sseli 3244	. . 3 ⊢ (𝐴 ∈ P → 𝐴 ∈ (𝒫 Q × 𝒫 Q))
3		1st2nd2 6403	. . 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)

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209  𝒫 cpw 3688  ⟨cop 3711   × cxp 4770  ‘cfv 5375  1^st c1st 6366  2^nd c2nd 6367  Qcnq 7641  Pcnp 7652

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 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

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 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-iota 5335  df-fun 5377  df-fv 5383  df-1st 6368  df-2nd 6369  df-inp 7827

This theorem is referenced by:  elnp1st2nd  7837  0npr  7844  genpdf  7869  genipv  7870  genpelvl  7873  genpelvu  7874  genpml  7878  genpmu  7879  genprndl  7882  genprndu  7883  genpdisj  7884  genpassl  7885  genpassu  7886  addnqprl  7890  addnqpru  7891  addlocprlemeqgt  7893  addlocprlemgt  7895  addlocprlem  7896  addlocpr  7897  nqprl  7912  nqpru  7913  addnqprlemfl  7920  addnqprlemfu  7921  mulnqprl  7929  mulnqpru  7930  mullocprlem  7931  mullocpr  7932  mulnqprlemfl  7936  mulnqprlemfu  7937  addcomprg  7939  mulcomprg  7941  distrlem1prl  7943  distrlem1pru  7944  distrlem4prl  7945  distrlem4pru  7946  ltprordil  7950  1idprl  7951  1idpru  7952  ltpopr  7956  ltsopr  7957  ltaddpr  7958  ltexprlemm  7961  ltexprlemopl  7962  ltexprlemlol  7963  ltexprlemopu  7964  ltexprlemupu  7965  ltexprlemdisj  7967  ltexprlemloc  7968  ltexprlemfl  7970  ltexprlemrl  7971  ltexprlemfu  7972  ltexprlemru  7973  addcanprleml  7975  addcanprlemu  7976  prplnqu  7981  recexprlemm  7985  recexprlemdisj  7991  recexprlemloc  7992  recexprlem1ssl  7994  recexprlem1ssu  7995  recexprlemss1l  7996  recexprlemss1u  7997  aptiprleml  8000  aptiprlemu  8001  archpr  8004  cauappcvgprlemladdru  8017  cauappcvgprlemladdrl  8018  archrecpr  8025  caucvgprlemladdrl  8039  caucvgprprlemml  8055  caucvgprprlemmu  8056  caucvgprprlemopl  8058  suplocexprlemml  8077  suplocexprlemrl  8078  suplocexprlemmu  8079  suplocexprlemdisj  8081  suplocexprlemloc  8082  suplocexprlemex  8083  suplocexprlemub  8084