MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elnpi Structured version   Visualization version   GIF version

Theorem elnpi 10097
Description: Membership in positive reals. (Contributed by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
elnpi (𝐴P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem elnpi
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elex 3399 . 2 (𝐴P𝐴 ∈ V)
2 simpl1 1243 . 2 (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)) → 𝐴 ∈ V)
3 psseq2 3891 . . . . . 6 (𝑧 = 𝐴 → (∅ ⊊ 𝑧 ↔ ∅ ⊊ 𝐴))
4 psseq1 3890 . . . . . 6 (𝑧 = 𝐴 → (𝑧Q𝐴Q))
53, 4anbi12d 625 . . . . 5 (𝑧 = 𝐴 → ((∅ ⊊ 𝑧𝑧Q) ↔ (∅ ⊊ 𝐴𝐴Q)))
6 eleq2 2866 . . . . . . . . 9 (𝑧 = 𝐴 → (𝑦𝑧𝑦𝐴))
76imbi2d 332 . . . . . . . 8 (𝑧 = 𝐴 → ((𝑦 <Q 𝑥𝑦𝑧) ↔ (𝑦 <Q 𝑥𝑦𝐴)))
87albidv 2016 . . . . . . 7 (𝑧 = 𝐴 → (∀𝑦(𝑦 <Q 𝑥𝑦𝑧) ↔ ∀𝑦(𝑦 <Q 𝑥𝑦𝐴)))
9 rexeq 3321 . . . . . . 7 (𝑧 = 𝐴 → (∃𝑦𝑧 𝑥 <Q 𝑦 ↔ ∃𝑦𝐴 𝑥 <Q 𝑦))
108, 9anbi12d 625 . . . . . 6 (𝑧 = 𝐴 → ((∀𝑦(𝑦 <Q 𝑥𝑦𝑧) ∧ ∃𝑦𝑧 𝑥 <Q 𝑦) ↔ (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
1110raleqbi1dv 3328 . . . . 5 (𝑧 = 𝐴 → (∀𝑥𝑧 (∀𝑦(𝑦 <Q 𝑥𝑦𝑧) ∧ ∃𝑦𝑧 𝑥 <Q 𝑦) ↔ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
125, 11anbi12d 625 . . . 4 (𝑧 = 𝐴 → (((∅ ⊊ 𝑧𝑧Q) ∧ ∀𝑥𝑧 (∀𝑦(𝑦 <Q 𝑥𝑦𝑧) ∧ ∃𝑦𝑧 𝑥 <Q 𝑦)) ↔ ((∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦))))
13 df-np 10090 . . . 4 P = {𝑧 ∣ ((∅ ⊊ 𝑧𝑧Q) ∧ ∀𝑥𝑧 (∀𝑦(𝑦 <Q 𝑥𝑦𝑧) ∧ ∃𝑦𝑧 𝑥 <Q 𝑦))}
1412, 13elab2g 3544 . . 3 (𝐴 ∈ V → (𝐴P ↔ ((∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦))))
15 id 22 . . . . . 6 ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) → (𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q))
16153expib 1153 . . . . 5 (𝐴 ∈ V → ((∅ ⊊ 𝐴𝐴Q) → (𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q)))
17 3simpc 1183 . . . . 5 ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) → (∅ ⊊ 𝐴𝐴Q))
1816, 17impbid1 217 . . . 4 (𝐴 ∈ V → ((∅ ⊊ 𝐴𝐴Q) ↔ (𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q)))
1918anbi1d 624 . . 3 (𝐴 ∈ V → (((∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)) ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦))))
2014, 19bitrd 271 . 2 (𝐴 ∈ V → (𝐴P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦))))
211, 2, 20pm5.21nii 370 1 (𝐴P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108  wal 1651   = wceq 1653  wcel 2157  wral 3088  wrex 3089  Vcvv 3384  wpss 3769  c0 4114   class class class wbr 4842  Qcnq 9961   <Q cltq 9967  Pcnp 9968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2776
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ne 2971  df-ral 3093  df-rex 3094  df-v 3386  df-in 3775  df-ss 3782  df-pss 3784  df-np 10090
This theorem is referenced by:  prn0  10098  prpssnq  10099  prcdnq  10102  prnmax  10104
  Copyright terms: Public domain W3C validator