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Theorem elwina 10102
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 3518 . 2 (𝐴 ∈ Inaccw𝐴 ∈ V)
2 fvex 6682 . . . 4 (cf‘𝐴) ∈ V
3 eleq1 2905 . . . 4 ((cf‘𝐴) = 𝐴 → ((cf‘𝐴) ∈ V ↔ 𝐴 ∈ V))
42, 3mpbii 234 . . 3 ((cf‘𝐴) = 𝐴𝐴 ∈ V)
543ad2ant2 1128 . 2 ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦) → 𝐴 ∈ V)
6 neeq1 3083 . . . 4 (𝑧 = 𝐴 → (𝑧 ≠ ∅ ↔ 𝐴 ≠ ∅))
7 fveq2 6669 . . . . 5 (𝑧 = 𝐴 → (cf‘𝑧) = (cf‘𝐴))
8 eqeq12 2840 . . . . 5 (((cf‘𝑧) = (cf‘𝐴) ∧ 𝑧 = 𝐴) → ((cf‘𝑧) = 𝑧 ↔ (cf‘𝐴) = 𝐴))
97, 8mpancom 684 . . . 4 (𝑧 = 𝐴 → ((cf‘𝑧) = 𝑧 ↔ (cf‘𝐴) = 𝐴))
10 rexeq 3412 . . . . 5 (𝑧 = 𝐴 → (∃𝑦𝑧 𝑥𝑦 ↔ ∃𝑦𝐴 𝑥𝑦))
1110raleqbi1dv 3409 . . . 4 (𝑧 = 𝐴 → (∀𝑥𝑧𝑦𝑧 𝑥𝑦 ↔ ∀𝑥𝐴𝑦𝐴 𝑥𝑦))
126, 9, 113anbi123d 1429 . . 3 (𝑧 = 𝐴 → ((𝑧 ≠ ∅ ∧ (cf‘𝑧) = 𝑧 ∧ ∀𝑥𝑧𝑦𝑧 𝑥𝑦) ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)))
13 df-wina 10100 . . 3 Inaccw = {𝑧 ∣ (𝑧 ≠ ∅ ∧ (cf‘𝑧) = 𝑧 ∧ ∀𝑥𝑧𝑦𝑧 𝑥𝑦)}
1412, 13elab2g 3673 . 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 207  w3a 1081   = wceq 1530  wcel 2107  wne 3021  wral 3143  wrex 3144  Vcvv 3500  c0 4295   class class class wbr 5063  cfv 6354  csdm 8502  cfccf 9360  Inaccwcwina 10098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-nul 5207
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-iota 6313  df-fv 6362  df-wina 10100
This theorem is referenced by:  winaon  10104  inawina  10106  winacard  10108  winainf  10110  winalim2  10112  winafp  10113  gchina  10115
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