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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-0nel1 | Structured version Visualization version GIF version |
Description: The empty set does not belong to {1o}. (Contributed by BJ, 6-Apr-2019.) |
Ref | Expression |
---|---|
bj-0nel1 | ⊢ ∅ ∉ {1o} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1n0 8150 | . . . 4 ⊢ 1o ≠ ∅ | |
2 | 1 | nesymi 2991 | . . 3 ⊢ ¬ ∅ = 1o |
3 | 0ex 5175 | . . . 4 ⊢ ∅ ∈ V | |
4 | 3 | elsn 4531 | . . 3 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o) |
5 | 2, 4 | mtbir 326 | . 2 ⊢ ¬ ∅ ∈ {1o} |
6 | 5 | nelir 3041 | 1 ⊢ ∅ ∉ {1o} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2114 ∉ wnel 3038 ∅c0 4211 {csn 4516 1oc1o 8124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ne 2935 df-nel 3039 df-v 3400 df-dif 3846 df-un 3848 df-nul 4212 df-sn 4517 df-suc 6178 df-1o 8131 |
This theorem is referenced by: (None) |
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