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Mirrors > Home > MPE Home > Th. List > 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 8438 | . . . 4 ⊢ 1o ≠ ∅ | |
2 | 1 | nesymi 2998 | . . 3 ⊢ ¬ ∅ = 1o |
3 | nsuceq0 6404 | . . . . 5 ⊢ suc 1o ≠ ∅ | |
4 | necom 2994 | . . . . . 6 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o) | |
5 | df-2o 8417 | . . . . . . 7 ⊢ 2o = suc 1o | |
6 | 5 | neeq2i 3006 | . . . . . 6 ⊢ (∅ ≠ 2o ↔ ∅ ≠ suc 1o) |
7 | 4, 6 | bitr4i 278 | . . . . 5 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ 2o) |
8 | 3, 7 | mpbi 229 | . . . 4 ⊢ ∅ ≠ 2o |
9 | 8 | neii 2942 | . . 3 ⊢ ¬ ∅ = 2o |
10 | 2, 9 | pm3.2ni 880 | . 2 ⊢ ¬ (∅ = 1o ∨ ∅ = 2o) |
11 | 0ex 5268 | . . 3 ⊢ ∅ ∈ V | |
12 | 11 | elpr 4613 | . 2 ⊢ (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o)) |
13 | 10, 12 | mtbir 323 | 1 ⊢ ¬ ∅ ∈ {1o, 2o} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∅c0 4286 {cpr 4592 suc csuc 6323 1oc1o 8409 2oc2o 8410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-nul 5267 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-v 3449 df-dif 3917 df-un 3919 df-nul 4287 df-sn 4591 df-pr 4593 df-suc 6327 df-1o 8416 df-2o 8417 |
This theorem is referenced by: nosgnn0i 27030 sltres 27033 noseponlem 27035 sltso 27047 nosepssdm 27057 nodenselem8 27062 nolt02olem 27065 |
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