Theorem prop 7806

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 7802	. . . 4 ⊢ P ⊆ (𝒫 Q × 𝒫 Q)
2	1	sseli 3238	. . 3 ⊢ (𝐴 ∈ P → 𝐴 ∈ (𝒫 Q × 𝒫 Q))
3		1st2nd2 6382	. . 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 2297	. . 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 1398   ∈ wcel 2205  𝒫 cpw 3674  ⟨cop 3697   × cxp 4752  ‘cfv 5357  1^st c1st 6345  2^nd c2nd 6346  Qcnq 7611  Pcnp 7622

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fv 5365  df-1st 6347  df-2nd 6348  df-inp 7797

This theorem is referenced by:  elnp1st2nd  7807  0npr  7814  genpdf  7839  genipv  7840  genpelvl  7843  genpelvu  7844  genpml  7848  genpmu  7849  genprndl  7852  genprndu  7853  genpdisj  7854  genpassl  7855  genpassu  7856  addnqprl  7860  addnqpru  7861  addlocprlemeqgt  7863  addlocprlemgt  7865  addlocprlem  7866  addlocpr  7867  nqprl  7882  nqpru  7883  addnqprlemfl  7890  addnqprlemfu  7891  mulnqprl  7899  mulnqpru  7900  mullocprlem  7901  mullocpr  7902  mulnqprlemfl  7906  mulnqprlemfu  7907  addcomprg  7909  mulcomprg  7911  distrlem1prl  7913  distrlem1pru  7914  distrlem4prl  7915  distrlem4pru  7916  ltprordil  7920  1idprl  7921  1idpru  7922  ltpopr  7926  ltsopr  7927  ltaddpr  7928  ltexprlemm  7931  ltexprlemopl  7932  ltexprlemlol  7933  ltexprlemopu  7934  ltexprlemupu  7935  ltexprlemdisj  7937  ltexprlemloc  7938  ltexprlemfl  7940  ltexprlemrl  7941  ltexprlemfu  7942  ltexprlemru  7943  addcanprleml  7945  addcanprlemu  7946  prplnqu  7951  recexprlemm  7955  recexprlemdisj  7961  recexprlemloc  7962  recexprlem1ssl  7964  recexprlem1ssu  7965  recexprlemss1l  7966  recexprlemss1u  7967  aptiprleml  7970  aptiprlemu  7971  archpr  7974  cauappcvgprlemladdru  7987  cauappcvgprlemladdrl  7988  archrecpr  7995  caucvgprlemladdrl  8009  caucvgprprlemml  8025  caucvgprprlemmu  8026  caucvgprprlemopl  8028  suplocexprlemml  8047  suplocexprlemrl  8048  suplocexprlemmu  8049  suplocexprlemdisj  8051  suplocexprlemloc  8052  suplocexprlemex  8053  suplocexprlemub  8054