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Mirrors > Home > MPE Home > Th. List > n0el | Structured version Visualization version GIF version |
Description: Negated membership of the empty set in another class. (Contributed by Rodolfo Medina, 25-Sep-2010.) |
Ref | Expression |
---|---|
n0el | ⊢ (¬ ∅ ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑢 𝑢 ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 3140 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥)) | |
2 | df-ex 1772 | . . 3 ⊢ (∃𝑢 𝑢 ∈ 𝑥 ↔ ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥) | |
3 | 2 | ralbii 3162 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑢 𝑢 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥) |
4 | alnex 1773 | . . 3 ⊢ (∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥)) | |
5 | imnang 1833 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥) ↔ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥)) | |
6 | 0el 4317 | . . . . 5 ⊢ (∅ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑢 ¬ 𝑢 ∈ 𝑥) | |
7 | df-rex 3141 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑢 ¬ 𝑢 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥)) | |
8 | 6, 7 | bitri 276 | . . . 4 ⊢ (∅ ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥)) |
9 | 8 | notbii 321 | . . 3 ⊢ (¬ ∅ ∈ 𝐴 ↔ ¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑢 ¬ 𝑢 ∈ 𝑥)) |
10 | 4, 5, 9 | 3bitr4ri 305 | . 2 ⊢ (¬ ∅ ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ ∀𝑢 ¬ 𝑢 ∈ 𝑥)) |
11 | 1, 3, 10 | 3bitr4ri 305 | 1 ⊢ (¬ ∅ ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑢 𝑢 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1526 ∃wex 1771 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 ∅c0 4288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-dif 3936 df-nul 4289 |
This theorem is referenced by: n0el2 35471 prter2 35897 |
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