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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 33598 | . . 3 ⊢ ¬ ∅ ∈ {1o, 2o} | |
2 | nosgnn0i.1 | . . . 4 ⊢ 𝑋 ∈ {1o, 2o} | |
3 | eleq1 2825 | . . . 4 ⊢ (∅ = 𝑋 → (∅ ∈ {1o, 2o} ↔ 𝑋 ∈ {1o, 2o})) | |
4 | 2, 3 | mpbiri 261 | . . 3 ⊢ (∅ = 𝑋 → ∅ ∈ {1o, 2o}) |
5 | 1, 4 | mto 200 | . 2 ⊢ ¬ ∅ = 𝑋 |
6 | 5 | neir 2943 | 1 ⊢ ∅ ≠ 𝑋 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ∅c0 4237 {cpr 4543 1oc1o 8195 2oc2o 8196 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-nul 5199 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3410 df-dif 3869 df-un 3871 df-nul 4238 df-sn 4542 df-pr 4544 df-suc 6219 df-1o 8202 df-2o 8203 |
This theorem is referenced by: sltres 33602 noextenddif 33608 nolesgn2ores 33612 nosepnelem 33619 nosepdmlem 33623 nolt02o 33635 nosupbnd1lem3 33650 nosupbnd1lem5 33652 nosupbnd2lem1 33655 |
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