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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 8102 | . . . 4 ⊢ 1o ≠ ∅ | |
2 | 1 | nesymi 3044 | . . 3 ⊢ ¬ ∅ = 1o |
3 | 0ex 5175 | . . . 4 ⊢ ∅ ∈ V | |
4 | 3 | elsn 4540 | . . 3 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o) |
5 | 2, 4 | mtbir 326 | . 2 ⊢ ¬ ∅ ∈ {1o} |
6 | 5 | nelir 3094 | 1 ⊢ ∅ ∉ {1o} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 ∉ wnel 3091 ∅c0 4243 {csn 4525 1oc1o 8078 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-nel 3092 df-v 3443 df-dif 3884 df-un 3886 df-nul 4244 df-sn 4526 df-suc 6165 df-1o 8085 |
This theorem is referenced by: (None) |
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