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| Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-issetft | Structured version Visualization version GIF version | ||
| Description: A closed form of issetf 3496. The proof here is a modification of a subproof in vtoclgft 3551, where it could be used to shorten the proof. (Contributed by Wolf Lammen, 25-Jan-2025.) | 
| Ref | Expression | 
|---|---|
| wl-issetft | ⊢ (Ⅎ𝑥𝐴 → (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | isset 3493 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴) | |
| 2 | nfv 1913 | . . . . 5 ⊢ Ⅎ𝑦Ⅎ𝑥𝐴 | |
| 3 | nfnfc1 2907 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | |
| 4 | nfcvd 2905 | . . . . . . 7 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝑦) | |
| 5 | id 22 | . . . . . . 7 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴) | |
| 6 | 4, 5 | nfeqd 2915 | . . . . . 6 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴) | 
| 7 | 6 | nfnd 1857 | . . . . 5 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 ¬ 𝑦 = 𝐴) | 
| 8 | nfvd 1914 | . . . . 5 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑦 ¬ 𝑥 = 𝐴) | |
| 9 | eqeq1 2740 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐴 ↔ 𝑥 = 𝐴)) | |
| 10 | 9 | notbid 318 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (¬ 𝑦 = 𝐴 ↔ ¬ 𝑥 = 𝐴)) | 
| 11 | 10 | a1i 11 | . . . . 5 ⊢ (Ⅎ𝑥𝐴 → (𝑦 = 𝑥 → (¬ 𝑦 = 𝐴 ↔ ¬ 𝑥 = 𝐴))) | 
| 12 | 2, 3, 7, 8, 11 | cbv2w 2338 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → (∀𝑦 ¬ 𝑦 = 𝐴 ↔ ∀𝑥 ¬ 𝑥 = 𝐴)) | 
| 13 | alnex 1780 | . . . 4 ⊢ (∀𝑦 ¬ 𝑦 = 𝐴 ↔ ¬ ∃𝑦 𝑦 = 𝐴) | |
| 14 | alnex 1780 | . . . 4 ⊢ (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴) | |
| 15 | 12, 13, 14 | 3bitr3g 313 | . . 3 ⊢ (Ⅎ𝑥𝐴 → (¬ ∃𝑦 𝑦 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴)) | 
| 16 | 15 | con4bid 317 | . 2 ⊢ (Ⅎ𝑥𝐴 → (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴)) | 
| 17 | 1, 16 | bitrid 283 | 1 ⊢ (Ⅎ𝑥𝐴 → (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∀wal 1537 = wceq 1539 ∃wex 1778 ∈ wcel 2107 Ⅎwnfc 2889 Vcvv 3479 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-v 3481 | 
| This theorem is referenced by: (None) | 
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