Users' Mathboxes Mathbox for Wolf Lammen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wl-issetft Structured version   Visualization version   GIF version

Theorem wl-issetft 35965
Description: A closed form of issetf 3457. The proof here is a modification of a subproof in vtoclgft 3507, where it could be used to shorten the proof. (Contributed by Wolf Lammen, 25-Jan-2025.)
Assertion
Ref Expression
wl-issetft (𝑥𝐴 → (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴))

Proof of Theorem wl-issetft
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 isset 3456 . 2 (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
2 nfv 1917 . . . . 5 𝑦𝑥𝐴
3 nfnfc1 2908 . . . . 5 𝑥𝑥𝐴
4 nfcvd 2906 . . . . . . 7 (𝑥𝐴𝑥𝑦)
5 id 22 . . . . . . 7 (𝑥𝐴𝑥𝐴)
64, 5nfeqd 2915 . . . . . 6 (𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
76nfnd 1861 . . . . 5 (𝑥𝐴 → Ⅎ𝑥 ¬ 𝑦 = 𝐴)
8 nfvd 1918 . . . . 5 (𝑥𝐴 → Ⅎ𝑦 ¬ 𝑥 = 𝐴)
9 eqeq1 2741 . . . . . . 7 (𝑦 = 𝑥 → (𝑦 = 𝐴𝑥 = 𝐴))
109notbid 317 . . . . . 6 (𝑦 = 𝑥 → (¬ 𝑦 = 𝐴 ↔ ¬ 𝑥 = 𝐴))
1110a1i 11 . . . . 5 (𝑥𝐴 → (𝑦 = 𝑥 → (¬ 𝑦 = 𝐴 ↔ ¬ 𝑥 = 𝐴)))
122, 3, 7, 8, 11cbv2w 2334 . . . 4 (𝑥𝐴 → (∀𝑦 ¬ 𝑦 = 𝐴 ↔ ∀𝑥 ¬ 𝑥 = 𝐴))
13 alnex 1783 . . . 4 (∀𝑦 ¬ 𝑦 = 𝐴 ↔ ¬ ∃𝑦 𝑦 = 𝐴)
14 alnex 1783 . . . 4 (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴)
1512, 13, 143bitr3g 312 . . 3 (𝑥𝐴 → (¬ ∃𝑦 𝑦 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴))
1615con4bid 316 . 2 (𝑥𝐴 → (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴))
171, 16bitrid 282 1 (𝑥𝐴 → (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wal 1539   = wceq 1541  wex 1781  wcel 2106  wnfc 2885  Vcvv 3443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-v 3445
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator