MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elwina Structured version   Visualization version   GIF version

Theorem elwina 9709
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 3361 . 2 (𝐴 ∈ Inaccw𝐴 ∈ V)
2 fvex 6342 . . . 4 (cf‘𝐴) ∈ V
3 eleq1 2837 . . . 4 ((cf‘𝐴) = 𝐴 → ((cf‘𝐴) ∈ V ↔ 𝐴 ∈ V))
42, 3mpbii 223 . . 3 ((cf‘𝐴) = 𝐴𝐴 ∈ V)
543ad2ant2 1127 . 2 ((𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦) → 𝐴 ∈ V)
6 neeq1 3004 . . . 4 (𝑧 = 𝐴 → (𝑧 ≠ ∅ ↔ 𝐴 ≠ ∅))
7 fveq2 6332 . . . . 5 (𝑧 = 𝐴 → (cf‘𝑧) = (cf‘𝐴))
8 eqeq12 2783 . . . . 5 (((cf‘𝑧) = (cf‘𝐴) ∧ 𝑧 = 𝐴) → ((cf‘𝑧) = 𝑧 ↔ (cf‘𝐴) = 𝐴))
97, 8mpancom 660 . . . 4 (𝑧 = 𝐴 → ((cf‘𝑧) = 𝑧 ↔ (cf‘𝐴) = 𝐴))
10 rexeq 3287 . . . . 5 (𝑧 = 𝐴 → (∃𝑦𝑧 𝑥𝑦 ↔ ∃𝑦𝐴 𝑥𝑦))
1110raleqbi1dv 3294 . . . 4 (𝑧 = 𝐴 → (∀𝑥𝑧𝑦𝑧 𝑥𝑦 ↔ ∀𝑥𝐴𝑦𝐴 𝑥𝑦))
126, 9, 113anbi123d 1546 . . 3 (𝑧 = 𝐴 → ((𝑧 ≠ ∅ ∧ (cf‘𝑧) = 𝑧 ∧ ∀𝑥𝑧𝑦𝑧 𝑥𝑦) ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)))
13 df-wina 9707 . . 3 Inaccw = {𝑧 ∣ (𝑧 ≠ ∅ ∧ (cf‘𝑧) = 𝑧 ∧ ∀𝑥𝑧𝑦𝑧 𝑥𝑦)}
1412, 13elab2g 3502 . 2 (𝐴 ∈ V → (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)))
151, 5, 14pm5.21nii 367 1 (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 196  w3a 1070   = wceq 1630  wcel 2144  wne 2942  wral 3060  wrex 3061  Vcvv 3349  c0 4061   class class class wbr 4784  cfv 6031  csdm 8107  cfccf 8962  Inaccwcwina 9705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-nul 4920
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-iota 5994  df-fv 6039  df-wina 9707
This theorem is referenced by:  winaon  9711  inawina  9713  winacard  9715  winainf  9717  winalim2  9719  winafp  9720  gchina  9722
  Copyright terms: Public domain W3C validator