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Theorem elwina 10676
Description: Conditions of weak inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.)
Assertion
Ref Expression
elwina (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦))
Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem elwina
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elex 3493 . 2 (𝐴 ∈ Inaccw𝐴 ∈ V)
2 fvex 6900 . . . 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 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦) → 𝐴 ∈ V)
6 neeq1 3004 . . . 4 (𝑧 = 𝐴 → (𝑧 ≠ ∅ ↔ 𝐴 ≠ ∅))
7 fveq2 6887 . . . . 5 (𝑧 = 𝐴 → (cf‘𝑧) = (cf‘𝐴))
8 eqeq12 2750 . . . . 5 (((cf‘𝑧) = (cf‘𝐴) ∧ 𝑧 = 𝐴) → ((cf‘𝑧) = 𝑧 ↔ (cf‘𝐴) = 𝐴))
97, 8mpancom 687 . . . 4 (𝑧 = 𝐴 → ((cf‘𝑧) = 𝑧 ↔ (cf‘𝐴) = 𝐴))
10 rexeq 3322 . . . . 5 (𝑧 = 𝐴 → (∃𝑦𝑧 𝑥𝑦 ↔ ∃𝑦𝐴 𝑥𝑦))
1110raleqbi1dv 3334 . . . 4 (𝑧 = 𝐴 → (∀𝑥𝑧𝑦𝑧 𝑥𝑦 ↔ ∀𝑥𝐴𝑦𝐴 𝑥𝑦))
126, 9, 113anbi123d 1437 . . 3 (𝑧 = 𝐴 → ((𝑧 ≠ ∅ ∧ (cf‘𝑧) = 𝑧 ∧ ∀𝑥𝑧𝑦𝑧 𝑥𝑦) ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)))
13 df-wina 10674 . . 3 Inaccw = {𝑧 ∣ (𝑧 ≠ ∅ ∧ (cf‘𝑧) = 𝑧 ∧ ∀𝑥𝑧𝑦𝑧 𝑥𝑦)}
1412, 13elab2g 3668 . 2 (𝐴 ∈ V → (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)))
151, 5, 14pm5.21nii 380 1 (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 205  w3a 1088   = wceq 1542  wcel 2107  wne 2941  wral 3062  wrex 3071  Vcvv 3475  c0 4320   class class class wbr 5146  cfv 6539  csdm 8933  cfccf 9927  Inaccwcwina 10672
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 5304
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 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-br 5147  df-iota 6491  df-fv 6547  df-wina 10674
This theorem is referenced by:  winaon  10678  inawina  10680  winacard  10682  winainf  10684  winalim2  10686  winafp  10687  gchina  10689
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