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| Mirrors > Home > MPE Home > Th. List > nosgnn0i | Structured version Visualization version GIF version | ||
| Description: If 𝑋 is a surreal sign, then it is not null. (Contributed by Scott Fenton, 3-Aug-2011.) | 
| Ref | Expression | 
|---|---|
| nosgnn0i.1 | ⊢ 𝑋 ∈ {1o, 2o} | 
| Ref | Expression | 
|---|---|
| nosgnn0i | ⊢ ∅ ≠ 𝑋 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nosgnn0 27703 | . . 3 ⊢ ¬ ∅ ∈ {1o, 2o} | |
| 2 | nosgnn0i.1 | . . . 4 ⊢ 𝑋 ∈ {1o, 2o} | |
| 3 | eleq1 2829 | . . . 4 ⊢ (∅ = 𝑋 → (∅ ∈ {1o, 2o} ↔ 𝑋 ∈ {1o, 2o})) | |
| 4 | 2, 3 | mpbiri 258 | . . 3 ⊢ (∅ = 𝑋 → ∅ ∈ {1o, 2o}) | 
| 5 | 1, 4 | mto 197 | . 2 ⊢ ¬ ∅ = 𝑋 | 
| 6 | 5 | neir 2943 | 1 ⊢ ∅ ≠ 𝑋 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 {cpr 4628 1oc1o 8499 2oc2o 8500 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-un 3956 df-nul 4334 df-sn 4627 df-pr 4629 df-suc 6390 df-1o 8506 df-2o 8507 | 
| This theorem is referenced by: sltres 27707 noextenddif 27713 nolesgn2ores 27717 nosepnelem 27724 nosepdmlem 27728 nolt02o 27740 nosupbnd1lem3 27755 nosupbnd1lem5 27757 nosupbnd2lem1 27760 | 
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