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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 7847 | . . . 4 ⊢ 1o ≠ ∅ | |
2 | 1 | nesymi 3056 | . . 3 ⊢ ¬ ∅ = 1o |
3 | 0ex 5016 | . . . 4 ⊢ ∅ ∈ V | |
4 | 3 | elsn 4414 | . . 3 ⊢ (∅ ∈ {1o} ↔ ∅ = 1o) |
5 | 2, 4 | mtbir 315 | . 2 ⊢ ¬ ∅ ∈ {1o} |
6 | 5 | nelir 3105 | 1 ⊢ ∅ ∉ {1o} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 ∈ wcel 2164 ∉ wnel 3102 ∅c0 4146 {csn 4399 1oc1o 7824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-nul 5015 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-v 3416 df-dif 3801 df-un 3803 df-nul 4147 df-sn 4400 df-suc 5973 df-1o 7831 |
This theorem is referenced by: (None) |
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