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Theorem elwina 10724
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 3499 . 2 (𝐴 ∈ Inaccw𝐴 ∈ V)
2 fvex 6920 . . . 4 (cf‘𝐴) ∈ V
3 eleq1 2827 . . . 4 ((cf‘𝐴) = 𝐴 → ((cf‘𝐴) ∈ V ↔ 𝐴 ∈ V))
42, 3mpbii 233 . . 3 ((cf‘𝐴) = 𝐴𝐴 ∈ V)
543ad2ant2 1133 . 2 ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦) → 𝐴 ∈ V)
6 neeq1 3001 . . . 4 (𝑧 = 𝐴 → (𝑧 ≠ ∅ ↔ 𝐴 ≠ ∅))
7 fveq2 6907 . . . . 5 (𝑧 = 𝐴 → (cf‘𝑧) = (cf‘𝐴))
8 eqeq12 2752 . . . . 5 (((cf‘𝑧) = (cf‘𝐴) ∧ 𝑧 = 𝐴) → ((cf‘𝑧) = 𝑧 ↔ (cf‘𝐴) = 𝐴))
97, 8mpancom 688 . . . 4 (𝑧 = 𝐴 → ((cf‘𝑧) = 𝑧 ↔ (cf‘𝐴) = 𝐴))
10 rexeq 3320 . . . . 5 (𝑧 = 𝐴 → (∃𝑦𝑧 𝑥𝑦 ↔ ∃𝑦𝐴 𝑥𝑦))
1110raleqbi1dv 3336 . . . 4 (𝑧 = 𝐴 → (∀𝑥𝑧𝑦𝑧 𝑥𝑦 ↔ ∀𝑥𝐴𝑦𝐴 𝑥𝑦))
126, 9, 113anbi123d 1435 . . 3 (𝑧 = 𝐴 → ((𝑧 ≠ ∅ ∧ (cf‘𝑧) = 𝑧 ∧ ∀𝑥𝑧𝑦𝑧 𝑥𝑦) ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)))
13 df-wina 10722 . . 3 Inaccw = {𝑧 ∣ (𝑧 ≠ ∅ ∧ (cf‘𝑧) = 𝑧 ∧ ∀𝑥𝑧𝑦𝑧 𝑥𝑦)}
1412, 13elab2g 3683 . 2 (𝐴 ∈ V → (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)))
151, 5, 14pm5.21nii 378 1 (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 206  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  c0 4339   class class class wbr 5148  cfv 6563  csdm 8983  cfccf 9975  Inaccwcwina 10720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-wina 10722
This theorem is referenced by:  winaon  10726  inawina  10728  winacard  10730  winainf  10732  winalim2  10734  winafp  10735  gchina  10737
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