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Theorem elina 9710
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 3361 . 2 (𝐴 ∈ Inacc → 𝐴 ∈ V)
2 fvex 6342 . . . 4 (cf‘𝐴) ∈ V
3 eleq1 2837 . . . 4 ((cf‘𝐴) = 𝐴 → ((cf‘𝐴) ∈ V ↔ 𝐴 ∈ V))
42, 3mpbii 223 . . 3 ((cf‘𝐴) = 𝐴𝐴 ∈ V)
543ad2ant2 1127 . 2 ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴) → 𝐴 ∈ V)
6 neeq1 3004 . . . 4 (𝑦 = 𝐴 → (𝑦 ≠ ∅ ↔ 𝐴 ≠ ∅))
7 fveq2 6332 . . . . 5 (𝑦 = 𝐴 → (cf‘𝑦) = (cf‘𝐴))
8 eqeq12 2783 . . . . 5 (((cf‘𝑦) = (cf‘𝐴) ∧ 𝑦 = 𝐴) → ((cf‘𝑦) = 𝑦 ↔ (cf‘𝐴) = 𝐴))
97, 8mpancom 660 . . . 4 (𝑦 = 𝐴 → ((cf‘𝑦) = 𝑦 ↔ (cf‘𝐴) = 𝐴))
10 breq2 4788 . . . . 5 (𝑦 = 𝐴 → (𝒫 𝑥𝑦 ↔ 𝒫 𝑥𝐴))
1110raleqbi1dv 3294 . . . 4 (𝑦 = 𝐴 → (∀𝑥𝑦 𝒫 𝑥𝑦 ↔ ∀𝑥𝐴 𝒫 𝑥𝐴))
126, 9, 113anbi123d 1546 . . 3 (𝑦 = 𝐴 → ((𝑦 ≠ ∅ ∧ (cf‘𝑦) = 𝑦 ∧ ∀𝑥𝑦 𝒫 𝑥𝑦) ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴)))
13 df-ina 9708 . . 3 Inacc = {𝑦 ∣ (𝑦 ≠ ∅ ∧ (cf‘𝑦) = 𝑦 ∧ ∀𝑥𝑦 𝒫 𝑥𝑦)}
1412, 13elab2g 3502 . 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 196  w3a 1070   = wceq 1630  wcel 2144  wne 2942  wral 3060  Vcvv 3349  c0 4061  𝒫 cpw 4295   class class class wbr 4784  cfv 6031  csdm 8107  cfccf 8962  Inacccina 9706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-nul 4920
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-iota 5994  df-fv 6039  df-ina 9708
This theorem is referenced by:  inawina  9713  omina  9714  gchina  9722  inar1  9798  inatsk  9801  tskcard  9804  tskuni  9806  gruina  9841  grur1  9843
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