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Theorem elina 9388
 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 3185 . 2 (𝐴 ∈ Inacc → 𝐴 ∈ V)
2 fvex 6113 . . . 4 (cf‘𝐴) ∈ V
3 eleq1 2676 . . . 4 ((cf‘𝐴) = 𝐴 → ((cf‘𝐴) ∈ V ↔ 𝐴 ∈ V))
42, 3mpbii 222 . . 3 ((cf‘𝐴) = 𝐴𝐴 ∈ V)
543ad2ant2 1076 . 2 ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴) → 𝐴 ∈ V)
6 neeq1 2844 . . . 4 (𝑦 = 𝐴 → (𝑦 ≠ ∅ ↔ 𝐴 ≠ ∅))
7 fveq2 6103 . . . . 5 (𝑦 = 𝐴 → (cf‘𝑦) = (cf‘𝐴))
8 eqeq12 2623 . . . . 5 (((cf‘𝑦) = (cf‘𝐴) ∧ 𝑦 = 𝐴) → ((cf‘𝑦) = 𝑦 ↔ (cf‘𝐴) = 𝐴))
97, 8mpancom 700 . . . 4 (𝑦 = 𝐴 → ((cf‘𝑦) = 𝑦 ↔ (cf‘𝐴) = 𝐴))
10 breq2 4587 . . . . 5 (𝑦 = 𝐴 → (𝒫 𝑥𝑦 ↔ 𝒫 𝑥𝐴))
1110raleqbi1dv 3123 . . . 4 (𝑦 = 𝐴 → (∀𝑥𝑦 𝒫 𝑥𝑦 ↔ ∀𝑥𝐴 𝒫 𝑥𝐴))
126, 9, 113anbi123d 1391 . . 3 (𝑦 = 𝐴 → ((𝑦 ≠ ∅ ∧ (cf‘𝑦) = 𝑦 ∧ ∀𝑥𝑦 𝒫 𝑥𝑦) ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴)))
13 df-ina 9386 . . 3 Inacc = {𝑦 ∣ (𝑦 ≠ ∅ ∧ (cf‘𝑦) = 𝑦 ∧ ∀𝑥𝑦 𝒫 𝑥𝑦)}
1412, 13elab2g 3322 . 2 (𝐴 ∈ V → (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴)))
151, 5, 14pm5.21nii 367 1 (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173  ∅c0 3874  𝒫 cpw 4108   class class class wbr 4583  ‘cfv 5804   ≺ csdm 7840  cfccf 8646  Inacccina 9384 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ina 9386 This theorem is referenced by:  inawina  9391  omina  9392  gchina  9400  inar1  9476  inatsk  9479  tskcard  9482  tskuni  9484  gruina  9519  grur1  9521
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