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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 3047 | . 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵) | |
2 | nfnel.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | nfnel.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
4 | 2, 3 | nfel 2917 | . . 3 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 |
5 | 4 | nfn 1860 | . 2 ⊢ Ⅎ𝑥 ¬ 𝐴 ∈ 𝐵 |
6 | 1, 5 | nfxfr 1855 | 1 ⊢ Ⅎ𝑥 𝐴 ∉ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 Ⅎwnf 1785 ∈ wcel 2106 Ⅎwnfc 2883 ∉ wnel 3046 |
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 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 df-cleq 2724 df-clel 2810 df-nfc 2885 df-nel 3047 |
This theorem is referenced by: (None) |
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