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Mathbox for Wolf Lammen |
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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 3495. The proof here is a modification of a subproof in vtoclgft 3552, 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 3492 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴) | |
2 | nfv 1912 | . . . . 5 ⊢ Ⅎ𝑦Ⅎ𝑥𝐴 | |
3 | nfnfc1 2906 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | |
4 | nfcvd 2904 | . . . . . . 7 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝑦) | |
5 | id 22 | . . . . . . 7 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴) | |
6 | 4, 5 | nfeqd 2914 | . . . . . 6 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴) |
7 | 6 | nfnd 1856 | . . . . 5 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 ¬ 𝑦 = 𝐴) |
8 | nfvd 1913 | . . . . 5 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑦 ¬ 𝑥 = 𝐴) | |
9 | eqeq1 2739 | . . . . . . 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 1778 | . . . 4 ⊢ (∀𝑦 ¬ 𝑦 = 𝐴 ↔ ¬ ∃𝑦 𝑦 = 𝐴) | |
14 | alnex 1778 | . . . 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 1535 = wceq 1537 ∃wex 1776 ∈ wcel 2106 Ⅎwnfc 2888 Vcvv 3478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-v 3480 |
This theorem is referenced by: (None) |
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