![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3039 | . 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵) | |
2 | nfnel.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | nfnel.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
4 | 2, 3 | nfel 2909 | . . 3 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 |
5 | 4 | nfn 1852 | . 2 ⊢ Ⅎ𝑥 ¬ 𝐴 ∈ 𝐵 |
6 | 1, 5 | nfxfr 1847 | 1 ⊢ Ⅎ𝑥 𝐴 ∉ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 Ⅎwnf 1777 ∈ wcel 2098 Ⅎwnfc 2875 ∉ wnel 3038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-cleq 2716 df-clel 2802 df-nfc 2877 df-nel 3039 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |