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Theorem elina 9944
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 3450 . 2 (𝐴 ∈ Inacc → 𝐴 ∈ V)
2 fvex 6543 . . . 4 (cf‘𝐴) ∈ V
3 eleq1 2868 . . . 4 ((cf‘𝐴) = 𝐴 → ((cf‘𝐴) ∈ V ↔ 𝐴 ∈ V))
42, 3mpbii 234 . . 3 ((cf‘𝐴) = 𝐴𝐴 ∈ V)
543ad2ant2 1125 . 2 ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴) → 𝐴 ∈ V)
6 neeq1 3044 . . . 4 (𝑦 = 𝐴 → (𝑦 ≠ ∅ ↔ 𝐴 ≠ ∅))
7 fveq2 6530 . . . . 5 (𝑦 = 𝐴 → (cf‘𝑦) = (cf‘𝐴))
8 eqeq12 2806 . . . . 5 (((cf‘𝑦) = (cf‘𝐴) ∧ 𝑦 = 𝐴) → ((cf‘𝑦) = 𝑦 ↔ (cf‘𝐴) = 𝐴))
97, 8mpancom 684 . . . 4 (𝑦 = 𝐴 → ((cf‘𝑦) = 𝑦 ↔ (cf‘𝐴) = 𝐴))
10 breq2 4960 . . . . 5 (𝑦 = 𝐴 → (𝒫 𝑥𝑦 ↔ 𝒫 𝑥𝐴))
1110raleqbi1dv 3360 . . . 4 (𝑦 = 𝐴 → (∀𝑥𝑦 𝒫 𝑥𝑦 ↔ ∀𝑥𝐴 𝒫 𝑥𝐴))
126, 9, 113anbi123d 1426 . . 3 (𝑦 = 𝐴 → ((𝑦 ≠ ∅ ∧ (cf‘𝑦) = 𝑦 ∧ ∀𝑥𝑦 𝒫 𝑥𝑦) ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴)))
13 df-ina 9942 . . 3 Inacc = {𝑦 ∣ (𝑦 ≠ ∅ ∧ (cf‘𝑦) = 𝑦 ∧ ∀𝑥𝑦 𝒫 𝑥𝑦)}
1412, 13elab2g 3602 . 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 207  w3a 1078   = wceq 1520  wcel 2079  wne 2982  wral 3103  Vcvv 3432  c0 4206  𝒫 cpw 4447   class class class wbr 4956  cfv 6217  csdm 8346  cfccf 9201  Inacccina 9940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-ext 2767  ax-nul 5095
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-sbc 3702  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-br 4957  df-iota 6181  df-fv 6225  df-ina 9942
This theorem is referenced by:  inawina  9947  omina  9948  gchina  9956  inar1  10032  inatsk  10035  tskcard  10038  tskuni  10040  gruina  10075  grur1  10077
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