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Mirrors > Home > MPE Home > Th. List > Mathboxes > nosgnn0 | Structured version Visualization version GIF version |
Description: ∅ is not a surreal sign. (Contributed by Scott Fenton, 16-Jun-2011.) |
Ref | Expression |
---|---|
nosgnn0 | ⊢ ¬ ∅ ∈ {1o, 2o} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1n0 8143 | . . . 4 ⊢ 1o ≠ ∅ | |
2 | 1 | nesymi 2991 | . . 3 ⊢ ¬ ∅ = 1o |
3 | nsuceq0 6246 | . . . . 5 ⊢ suc 1o ≠ ∅ | |
4 | necom 2987 | . . . . . 6 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o) | |
5 | df-2o 8125 | . . . . . . 7 ⊢ 2o = suc 1o | |
6 | 5 | neeq2i 2999 | . . . . . 6 ⊢ (∅ ≠ 2o ↔ ∅ ≠ suc 1o) |
7 | 4, 6 | bitr4i 281 | . . . . 5 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ 2o) |
8 | 3, 7 | mpbi 233 | . . . 4 ⊢ ∅ ≠ 2o |
9 | 8 | neii 2936 | . . 3 ⊢ ¬ ∅ = 2o |
10 | 2, 9 | pm3.2ni 880 | . 2 ⊢ ¬ (∅ = 1o ∨ ∅ = 2o) |
11 | 0ex 5172 | . . 3 ⊢ ∅ ∈ V | |
12 | 11 | elpr 4536 | . 2 ⊢ (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o)) |
13 | 10, 12 | mtbir 326 | 1 ⊢ ¬ ∅ ∈ {1o, 2o} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 846 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 ∅c0 4209 {cpr 4515 suc csuc 6168 1oc1o 8117 2oc2o 8118 |
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 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-ext 2710 ax-nul 5171 |
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 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-ne 2935 df-v 3399 df-dif 3844 df-un 3846 df-nul 4210 df-sn 4514 df-pr 4516 df-suc 6172 df-1o 8124 df-2o 8125 |
This theorem is referenced by: nosgnn0i 33495 sltres 33498 noseponlem 33500 sltso 33512 nosepssdm 33522 nodenselem8 33527 nolt02olem 33530 |
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