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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 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.) |
Ref | Expression |
---|---|
wl-issetft | ⊢ (Ⅎ𝑥𝐴 → (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isset 3456 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴) | |
2 | nfv 1917 | . . . . 5 ⊢ Ⅎ𝑦Ⅎ𝑥𝐴 | |
3 | nfnfc1 2908 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | |
4 | nfcvd 2906 | . . . . . . 7 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝑦) | |
5 | id 22 | . . . . . . 7 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴) | |
6 | 4, 5 | nfeqd 2915 | . . . . . 6 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴) |
7 | 6 | nfnd 1861 | . . . . 5 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 ¬ 𝑦 = 𝐴) |
8 | nfvd 1918 | . . . . 5 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑦 ¬ 𝑥 = 𝐴) | |
9 | eqeq1 2741 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐴 ↔ 𝑥 = 𝐴)) | |
10 | 9 | notbid 317 | . . . . . 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 1783 | . . . 4 ⊢ (∀𝑦 ¬ 𝑦 = 𝐴 ↔ ¬ ∃𝑦 𝑦 = 𝐴) | |
14 | alnex 1783 | . . . 4 ⊢ (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴) | |
15 | 12, 13, 14 | 3bitr3g 312 | . . 3 ⊢ (Ⅎ𝑥𝐴 → (¬ ∃𝑦 𝑦 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴)) |
16 | 15 | con4bid 316 | . 2 ⊢ (Ⅎ𝑥𝐴 → (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴)) |
17 | 1, 16 | bitrid 282 | 1 ⊢ (Ⅎ𝑥𝐴 → (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 Ⅎwnfc 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) |
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