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Theorem elnpi 11013
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 3480 . 2 (𝐴P𝐴 ∈ V)
2 simpl1 1188 . 2 (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)) → 𝐴 ∈ V)
3 psseq2 4084 . . . . . 6 (𝑧 = 𝐴 → (∅ ⊊ 𝑧 ↔ ∅ ⊊ 𝐴))
4 psseq1 4083 . . . . . 6 (𝑧 = 𝐴 → (𝑧Q𝐴Q))
53, 4anbi12d 630 . . . . 5 (𝑧 = 𝐴 → ((∅ ⊊ 𝑧𝑧Q) ↔ (∅ ⊊ 𝐴𝐴Q)))
6 eleq2 2814 . . . . . . . . 9 (𝑧 = 𝐴 → (𝑦𝑧𝑦𝐴))
76imbi2d 339 . . . . . . . 8 (𝑧 = 𝐴 → ((𝑦 <Q 𝑥𝑦𝑧) ↔ (𝑦 <Q 𝑥𝑦𝐴)))
87albidv 1915 . . . . . . 7 (𝑧 = 𝐴 → (∀𝑦(𝑦 <Q 𝑥𝑦𝑧) ↔ ∀𝑦(𝑦 <Q 𝑥𝑦𝐴)))
9 rexeq 3310 . . . . . . 7 (𝑧 = 𝐴 → (∃𝑦𝑧 𝑥 <Q 𝑦 ↔ ∃𝑦𝐴 𝑥 <Q 𝑦))
108, 9anbi12d 630 . . . . . 6 (𝑧 = 𝐴 → ((∀𝑦(𝑦 <Q 𝑥𝑦𝑧) ∧ ∃𝑦𝑧 𝑥 <Q 𝑦) ↔ (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
1110raleqbi1dv 3322 . . . . 5 (𝑧 = 𝐴 → (∀𝑥𝑧 (∀𝑦(𝑦 <Q 𝑥𝑦𝑧) ∧ ∃𝑦𝑧 𝑥 <Q 𝑦) ↔ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
125, 11anbi12d 630 . . . 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 11006 . . . 4 P = {𝑧 ∣ ((∅ ⊊ 𝑧𝑧Q) ∧ ∀𝑥𝑧 (∀𝑦(𝑦 <Q 𝑥𝑦𝑧) ∧ ∃𝑦𝑧 𝑥 <Q 𝑦))}
1412, 13elab2g 3666 . . 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 1119 . . . . 5 (𝐴 ∈ V → ((∅ ⊊ 𝐴𝐴Q) → (𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q)))
17 3simpc 1147 . . . . 5 ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) → (∅ ⊊ 𝐴𝐴Q))
1816, 17impbid1 224 . . . 4 (𝐴 ∈ V → ((∅ ⊊ 𝐴𝐴Q) ↔ (𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q)))
1918anbi1d 629 . . 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 278 . 2 (𝐴 ∈ V → (𝐴P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦))))
211, 2, 20pm5.21nii 377 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 205  wa 394  w3a 1084  wal 1531   = wceq 1533  wcel 2098  wral 3050  wrex 3059  Vcvv 3461  wpss 3945  c0 4322   class class class wbr 5149  Qcnq 10877   <Q cltq 10883  Pcnp 10884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-v 3463  df-ss 3961  df-pss 3964  df-np 11006
This theorem is referenced by:  prn0  11014  prpssnq  11015  prcdnq  11018  prnmax  11020
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