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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 8483 | . . . 4 ⊢ 1o ≠ ∅ | |
2 | 1 | nesymi 2990 | . . 3 ⊢ ¬ ∅ = 1o |
3 | 0ex 5297 | . . . 4 ⊢ ∅ ∈ V | |
4 | 3 | elsn 4635 | . . 3 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o) |
5 | 2, 4 | mtbir 323 | . 2 ⊢ ¬ ∅ ∈ {1o} |
6 | 5 | nelir 3041 | 1 ⊢ ∅ ∉ {1o} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ∉ wnel 3038 ∅c0 4314 {csn 4620 1oc1o 8454 |
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-ext 2695 ax-nul 5296 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-nel 3039 df-v 3468 df-dif 3943 df-un 3945 df-nul 4315 df-sn 4621 df-suc 6360 df-1o 8461 |
This theorem is referenced by: (None) |
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