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Theorem elina 9494
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 3207 . 2 (𝐴 ∈ Inacc → 𝐴 ∈ V)
2 fvex 6188 . . . 4 (cf‘𝐴) ∈ V
3 eleq1 2687 . . . 4 ((cf‘𝐴) = 𝐴 → ((cf‘𝐴) ∈ V ↔ 𝐴 ∈ V))
42, 3mpbii 223 . . 3 ((cf‘𝐴) = 𝐴𝐴 ∈ V)
543ad2ant2 1081 . 2 ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴) → 𝐴 ∈ V)
6 neeq1 2853 . . . 4 (𝑦 = 𝐴 → (𝑦 ≠ ∅ ↔ 𝐴 ≠ ∅))
7 fveq2 6178 . . . . 5 (𝑦 = 𝐴 → (cf‘𝑦) = (cf‘𝐴))
8 eqeq12 2633 . . . . 5 (((cf‘𝑦) = (cf‘𝐴) ∧ 𝑦 = 𝐴) → ((cf‘𝑦) = 𝑦 ↔ (cf‘𝐴) = 𝐴))
97, 8mpancom 702 . . . 4 (𝑦 = 𝐴 → ((cf‘𝑦) = 𝑦 ↔ (cf‘𝐴) = 𝐴))
10 breq2 4648 . . . . 5 (𝑦 = 𝐴 → (𝒫 𝑥𝑦 ↔ 𝒫 𝑥𝐴))
1110raleqbi1dv 3141 . . . 4 (𝑦 = 𝐴 → (∀𝑥𝑦 𝒫 𝑥𝑦 ↔ ∀𝑥𝐴 𝒫 𝑥𝐴))
126, 9, 113anbi123d 1397 . . 3 (𝑦 = 𝐴 → ((𝑦 ≠ ∅ ∧ (cf‘𝑦) = 𝑦 ∧ ∀𝑥𝑦 𝒫 𝑥𝑦) ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴)))
13 df-ina 9492 . . 3 Inacc = {𝑦 ∣ (𝑦 ≠ ∅ ∧ (cf‘𝑦) = 𝑦 ∧ ∀𝑥𝑦 𝒫 𝑥𝑦)}
1412, 13elab2g 3347 . 2 (𝐴 ∈ V → (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴)))
151, 5, 14pm5.21nii 368 1 (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 196  w3a 1036   = wceq 1481  wcel 1988  wne 2791  wral 2909  Vcvv 3195  c0 3907  𝒫 cpw 4149   class class class wbr 4644  cfv 5876  csdm 7939  cfccf 8748  Inacccina 9490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-nul 4780
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-ina 9492
This theorem is referenced by:  inawina  9497  omina  9498  gchina  9506  inar1  9582  inatsk  9585  tskcard  9588  tskuni  9590  gruina  9625  grur1  9627
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