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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 3462. The proof here is a modification of a subproof in vtoclgft 3512, 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 3461 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴) | |
2 | nfv 1918 | . . . . 5 ⊢ Ⅎ𝑦Ⅎ𝑥𝐴 | |
3 | nfnfc1 2911 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | |
4 | nfcvd 2909 | . . . . . . 7 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝑦) | |
5 | id 22 | . . . . . . 7 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴) | |
6 | 4, 5 | nfeqd 2918 | . . . . . 6 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴) |
7 | 6 | nfnd 1862 | . . . . 5 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 ¬ 𝑦 = 𝐴) |
8 | nfvd 1919 | . . . . 5 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑦 ¬ 𝑥 = 𝐴) | |
9 | eqeq1 2741 | . . . . . . 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 2334 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → (∀𝑦 ¬ 𝑦 = 𝐴 ↔ ∀𝑥 ¬ 𝑥 = 𝐴)) |
13 | alnex 1784 | . . . 4 ⊢ (∀𝑦 ¬ 𝑦 = 𝐴 ↔ ¬ ∃𝑦 𝑦 = 𝐴) | |
14 | alnex 1784 | . . . 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 205 ∀wal 1540 = wceq 1542 ∃wex 1782 ∈ wcel 2107 Ⅎwnfc 2888 Vcvv 3448 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-v 3450 |
This theorem is referenced by: (None) |
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