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Theorem elnpi 9666
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 3184 . 2 (𝐴P𝐴 ∈ V)
2 simpl1 1056 . 2 (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)) → 𝐴 ∈ V)
3 psseq2 3656 . . . . . 6 (𝑧 = 𝐴 → (∅ ⊊ 𝑧 ↔ ∅ ⊊ 𝐴))
4 psseq1 3655 . . . . . 6 (𝑧 = 𝐴 → (𝑧Q𝐴Q))
53, 4anbi12d 742 . . . . 5 (𝑧 = 𝐴 → ((∅ ⊊ 𝑧𝑧Q) ↔ (∅ ⊊ 𝐴𝐴Q)))
6 eleq2 2676 . . . . . . . . 9 (𝑧 = 𝐴 → (𝑦𝑧𝑦𝐴))
76imbi2d 328 . . . . . . . 8 (𝑧 = 𝐴 → ((𝑦 <Q 𝑥𝑦𝑧) ↔ (𝑦 <Q 𝑥𝑦𝐴)))
87albidv 1835 . . . . . . 7 (𝑧 = 𝐴 → (∀𝑦(𝑦 <Q 𝑥𝑦𝑧) ↔ ∀𝑦(𝑦 <Q 𝑥𝑦𝐴)))
9 rexeq 3115 . . . . . . 7 (𝑧 = 𝐴 → (∃𝑦𝑧 𝑥 <Q 𝑦 ↔ ∃𝑦𝐴 𝑥 <Q 𝑦))
108, 9anbi12d 742 . . . . . 6 (𝑧 = 𝐴 → ((∀𝑦(𝑦 <Q 𝑥𝑦𝑧) ∧ ∃𝑦𝑧 𝑥 <Q 𝑦) ↔ (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
1110raleqbi1dv 3122 . . . . 5 (𝑧 = 𝐴 → (∀𝑥𝑧 (∀𝑦(𝑦 <Q 𝑥𝑦𝑧) ∧ ∃𝑦𝑧 𝑥 <Q 𝑦) ↔ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
125, 11anbi12d 742 . . . 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 9659 . . . 4 P = {𝑧 ∣ ((∅ ⊊ 𝑧𝑧Q) ∧ ∀𝑥𝑧 (∀𝑦(𝑦 <Q 𝑥𝑦𝑧) ∧ ∃𝑦𝑧 𝑥 <Q 𝑦))}
1412, 13elab2g 3321 . . 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 1259 . . . . 5 (𝐴 ∈ V → ((∅ ⊊ 𝐴𝐴Q) → (𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q)))
17 3simpc 1052 . . . . 5 ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) → (∅ ⊊ 𝐴𝐴Q))
1816, 17impbid1 213 . . . 4 (𝐴 ∈ V → ((∅ ⊊ 𝐴𝐴Q) ↔ (𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q)))
1918anbi1d 736 . . 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 266 . 2 (𝐴 ∈ V → (𝐴P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦))))
211, 2, 20pm5.21nii 366 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 194  wa 382  w3a 1030  wal 1472   = wceq 1474  wcel 1976  wral 2895  wrex 2896  Vcvv 3172  wpss 3540  c0 3873   class class class wbr 4577  Qcnq 9530   <Q cltq 9536  Pcnp 9537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-v 3174  df-in 3546  df-ss 3553  df-pss 3555  df-np 9659
This theorem is referenced by:  prn0  9667  prpssnq  9668  prcdnq  9671  prnmax  9673
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