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Theorem elina 10682
Description: Conditions of strong inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.)
Assertion
Ref Expression
elina (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴))
Distinct variable group:   𝑥,𝐴

Proof of Theorem elina
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elex 3493 . 2 (𝐴 ∈ Inacc → 𝐴 ∈ V)
2 fvex 6905 . . . 4 (cf‘𝐴) ∈ V
3 eleq1 2822 . . . 4 ((cf‘𝐴) = 𝐴 → ((cf‘𝐴) ∈ V ↔ 𝐴 ∈ V))
42, 3mpbii 232 . . 3 ((cf‘𝐴) = 𝐴𝐴 ∈ V)
543ad2ant2 1135 . 2 ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴) → 𝐴 ∈ V)
6 neeq1 3004 . . . 4 (𝑦 = 𝐴 → (𝑦 ≠ ∅ ↔ 𝐴 ≠ ∅))
7 fveq2 6892 . . . . 5 (𝑦 = 𝐴 → (cf‘𝑦) = (cf‘𝐴))
8 eqeq12 2750 . . . . 5 (((cf‘𝑦) = (cf‘𝐴) ∧ 𝑦 = 𝐴) → ((cf‘𝑦) = 𝑦 ↔ (cf‘𝐴) = 𝐴))
97, 8mpancom 687 . . . 4 (𝑦 = 𝐴 → ((cf‘𝑦) = 𝑦 ↔ (cf‘𝐴) = 𝐴))
10 breq2 5153 . . . . 5 (𝑦 = 𝐴 → (𝒫 𝑥𝑦 ↔ 𝒫 𝑥𝐴))
1110raleqbi1dv 3334 . . . 4 (𝑦 = 𝐴 → (∀𝑥𝑦 𝒫 𝑥𝑦 ↔ ∀𝑥𝐴 𝒫 𝑥𝐴))
126, 9, 113anbi123d 1437 . . 3 (𝑦 = 𝐴 → ((𝑦 ≠ ∅ ∧ (cf‘𝑦) = 𝑦 ∧ ∀𝑥𝑦 𝒫 𝑥𝑦) ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴)))
13 df-ina 10680 . . 3 Inacc = {𝑦 ∣ (𝑦 ≠ ∅ ∧ (cf‘𝑦) = 𝑦 ∧ ∀𝑥𝑦 𝒫 𝑥𝑦)}
1412, 13elab2g 3671 . 2 (𝐴 ∈ V → (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴)))
151, 5, 14pm5.21nii 380 1 (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 205  w3a 1088   = wceq 1542  wcel 2107  wne 2941  wral 3062  Vcvv 3475  c0 4323  𝒫 cpw 4603   class class class wbr 5149  cfv 6544  csdm 8938  cfccf 9932  Inacccina 10678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ina 10680
This theorem is referenced by:  inawina  10685  omina  10686  gchina  10694  inar1  10770  inatsk  10773  tskcard  10776  tskuni  10778  gruina  10813  grur1  10815
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