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Theorem elwina 9493
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 3207 . 2 (𝐴 ∈ Inaccw𝐴 ∈ 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 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦) → 𝐴 ∈ V)
6 neeq1 2853 . . . 4 (𝑧 = 𝐴 → (𝑧 ≠ ∅ ↔ 𝐴 ≠ ∅))
7 fveq2 6178 . . . . 5 (𝑧 = 𝐴 → (cf‘𝑧) = (cf‘𝐴))
8 eqeq12 2633 . . . . 5 (((cf‘𝑧) = (cf‘𝐴) ∧ 𝑧 = 𝐴) → ((cf‘𝑧) = 𝑧 ↔ (cf‘𝐴) = 𝐴))
97, 8mpancom 702 . . . 4 (𝑧 = 𝐴 → ((cf‘𝑧) = 𝑧 ↔ (cf‘𝐴) = 𝐴))
10 rexeq 3134 . . . . 5 (𝑧 = 𝐴 → (∃𝑦𝑧 𝑥𝑦 ↔ ∃𝑦𝐴 𝑥𝑦))
1110raleqbi1dv 3141 . . . 4 (𝑧 = 𝐴 → (∀𝑥𝑧𝑦𝑧 𝑥𝑦 ↔ ∀𝑥𝐴𝑦𝐴 𝑥𝑦))
126, 9, 113anbi123d 1397 . . 3 (𝑧 = 𝐴 → ((𝑧 ≠ ∅ ∧ (cf‘𝑧) = 𝑧 ∧ ∀𝑥𝑧𝑦𝑧 𝑥𝑦) ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)))
13 df-wina 9491 . . 3 Inaccw = {𝑧 ∣ (𝑧 ≠ ∅ ∧ (cf‘𝑧) = 𝑧 ∧ ∀𝑥𝑧𝑦𝑧 𝑥𝑦)}
1412, 13elab2g 3347 . 2 (𝐴 ∈ V → (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)))
151, 5, 14pm5.21nii 368 1 (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 196  w3a 1036   = wceq 1481  wcel 1988  wne 2791  wral 2909  wrex 2910  Vcvv 3195  c0 3907   class class class wbr 4644  cfv 5876  csdm 7939  cfccf 8748  Inaccwcwina 9489
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-wina 9491
This theorem is referenced by:  winaon  9495  inawina  9497  winacard  9499  winainf  9501  winalim2  9503  winafp  9504  gchina  9506
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