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Theorem elnpi 9923
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 3316 . 2 (𝐴P𝐴 ∈ V)
2 simpl1 1204 . 2 (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)) → 𝐴 ∈ V)
3 psseq2 3802 . . . . . 6 (𝑧 = 𝐴 → (∅ ⊊ 𝑧 ↔ ∅ ⊊ 𝐴))
4 psseq1 3801 . . . . . 6 (𝑧 = 𝐴 → (𝑧Q𝐴Q))
53, 4anbi12d 749 . . . . 5 (𝑧 = 𝐴 → ((∅ ⊊ 𝑧𝑧Q) ↔ (∅ ⊊ 𝐴𝐴Q)))
6 eleq2 2792 . . . . . . . . 9 (𝑧 = 𝐴 → (𝑦𝑧𝑦𝐴))
76imbi2d 329 . . . . . . . 8 (𝑧 = 𝐴 → ((𝑦 <Q 𝑥𝑦𝑧) ↔ (𝑦 <Q 𝑥𝑦𝐴)))
87albidv 1962 . . . . . . 7 (𝑧 = 𝐴 → (∀𝑦(𝑦 <Q 𝑥𝑦𝑧) ↔ ∀𝑦(𝑦 <Q 𝑥𝑦𝐴)))
9 rexeq 3242 . . . . . . 7 (𝑧 = 𝐴 → (∃𝑦𝑧 𝑥 <Q 𝑦 ↔ ∃𝑦𝐴 𝑥 <Q 𝑦))
108, 9anbi12d 749 . . . . . 6 (𝑧 = 𝐴 → ((∀𝑦(𝑦 <Q 𝑥𝑦𝑧) ∧ ∃𝑦𝑧 𝑥 <Q 𝑦) ↔ (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
1110raleqbi1dv 3249 . . . . 5 (𝑧 = 𝐴 → (∀𝑥𝑧 (∀𝑦(𝑦 <Q 𝑥𝑦𝑧) ∧ ∃𝑦𝑧 𝑥 <Q 𝑦) ↔ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
125, 11anbi12d 749 . . . 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 9916 . . . 4 P = {𝑧 ∣ ((∅ ⊊ 𝑧𝑧Q) ∧ ∀𝑥𝑧 (∀𝑦(𝑦 <Q 𝑥𝑦𝑧) ∧ ∃𝑦𝑧 𝑥 <Q 𝑦))}
1412, 13elab2g 3458 . . 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 1116 . . . . 5 (𝐴 ∈ V → ((∅ ⊊ 𝐴𝐴Q) → (𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q)))
17 3simpc 1144 . . . . 5 ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) → (∅ ⊊ 𝐴𝐴Q))
1816, 17impbid1 215 . . . 4 (𝐴 ∈ V → ((∅ ⊊ 𝐴𝐴Q) ↔ (𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q)))
1918anbi1d 743 . . 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 268 . 2 (𝐴 ∈ V → (𝐴P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦))))
211, 2, 20pm5.21nii 367 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 196  wa 383  w3a 1072  wal 1594   = wceq 1596  wcel 2103  wral 3014  wrex 3015  Vcvv 3304  wpss 3681  c0 4023   class class class wbr 4760  Qcnq 9787   <Q cltq 9793  Pcnp 9794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-v 3306  df-in 3687  df-ss 3694  df-pss 3696  df-np 9916
This theorem is referenced by:  prn0  9924  prpssnq  9925  prcdnq  9928  prnmax  9930
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