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| Mirrors > Home > MPE Home > Th. List > nfnid | Structured version Visualization version GIF version | ||
| Description: A setvar variable is not free from itself. This theorem is not true in a one-element domain, as illustrated by the use of dtruALT2 5370 in its proof. (Contributed by Mario Carneiro, 8-Oct-2016.) |
| Ref | Expression |
|---|---|
| nfnid | ⊢ ¬ Ⅎ𝑥𝑥 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dtruALT2 5370 | . . 3 ⊢ ¬ ∀𝑧 𝑧 = 𝑤 | |
| 2 | ax-ext 2708 | . . . . 5 ⊢ (∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑤) → 𝑧 = 𝑤) | |
| 3 | 2 | sps 2185 | . . . 4 ⊢ (∀𝑤∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑤) → 𝑧 = 𝑤) |
| 4 | 3 | alimi 1811 | . . 3 ⊢ (∀𝑧∀𝑤∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑤) → ∀𝑧 𝑧 = 𝑤) |
| 5 | 1, 4 | mto 197 | . 2 ⊢ ¬ ∀𝑧∀𝑤∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑤) |
| 6 | df-nfc 2892 | . . 3 ⊢ (Ⅎ𝑥𝑥 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝑥) | |
| 7 | sbnf2 2361 | . . . . 5 ⊢ (Ⅎ𝑥 𝑦 ∈ 𝑥 ↔ ∀𝑧∀𝑤([𝑧 / 𝑥]𝑦 ∈ 𝑥 ↔ [𝑤 / 𝑥]𝑦 ∈ 𝑥)) | |
| 8 | elsb2 2125 | . . . . . . 7 ⊢ ([𝑧 / 𝑥]𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧) | |
| 9 | elsb2 2125 | . . . . . . 7 ⊢ ([𝑤 / 𝑥]𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑤) | |
| 10 | 8, 9 | bibi12i 339 | . . . . . 6 ⊢ (([𝑧 / 𝑥]𝑦 ∈ 𝑥 ↔ [𝑤 / 𝑥]𝑦 ∈ 𝑥) ↔ (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑤)) |
| 11 | 10 | 2albii 1820 | . . . . 5 ⊢ (∀𝑧∀𝑤([𝑧 / 𝑥]𝑦 ∈ 𝑥 ↔ [𝑤 / 𝑥]𝑦 ∈ 𝑥) ↔ ∀𝑧∀𝑤(𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑤)) |
| 12 | 7, 11 | bitri 275 | . . . 4 ⊢ (Ⅎ𝑥 𝑦 ∈ 𝑥 ↔ ∀𝑧∀𝑤(𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑤)) |
| 13 | 12 | albii 1819 | . . 3 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝑥 ↔ ∀𝑦∀𝑧∀𝑤(𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑤)) |
| 14 | alrot3 2160 | . . 3 ⊢ (∀𝑦∀𝑧∀𝑤(𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑤) ↔ ∀𝑧∀𝑤∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑤)) | |
| 15 | 6, 13, 14 | 3bitri 297 | . 2 ⊢ (Ⅎ𝑥𝑥 ↔ ∀𝑧∀𝑤∀𝑦(𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑤)) |
| 16 | 5, 15 | mtbir 323 | 1 ⊢ ¬ Ⅎ𝑥𝑥 |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∀wal 1538 Ⅎwnf 1783 [wsb 2064 Ⅎwnfc 2890 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-nul 5306 ax-pow 5365 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-sb 2065 df-nfc 2892 |
| This theorem is referenced by: nfcvb 5376 |
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