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| Mirrors > Home > MPE Home > Th. List > nfnel | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.) | 
| Ref | Expression | 
|---|---|
| nfnel.1 | ⊢ Ⅎ𝑥𝐴 | 
| nfnel.2 | ⊢ Ⅎ𝑥𝐵 | 
| Ref | Expression | 
|---|---|
| nfnel | ⊢ Ⅎ𝑥 𝐴 ∉ 𝐵 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-nel 3046 | . 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵) | |
| 2 | nfnel.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 3 | nfnel.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
| 4 | 2, 3 | nfel 2919 | . . 3 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 | 
| 5 | 4 | nfn 1856 | . 2 ⊢ Ⅎ𝑥 ¬ 𝐴 ∈ 𝐵 | 
| 6 | 1, 5 | nfxfr 1852 | 1 ⊢ Ⅎ𝑥 𝐴 ∉ 𝐵 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 Ⅎwnf 1782 ∈ wcel 2107 Ⅎwnfc 2889 ∉ wnel 3045 | 
| 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-cleq 2728 df-clel 2815 df-nfc 2891 df-nel 3046 | 
| This theorem is referenced by: (None) | 
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