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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 33167 | . . 3 ⊢ ¬ ∅ ∈ {1o, 2o} | |
2 | nosgnn0i.1 | . . . 4 ⊢ 𝑋 ∈ {1o, 2o} | |
3 | eleq1 2902 | . . . 4 ⊢ (∅ = 𝑋 → (∅ ∈ {1o, 2o} ↔ 𝑋 ∈ {1o, 2o})) | |
4 | 2, 3 | mpbiri 260 | . . 3 ⊢ (∅ = 𝑋 → ∅ ∈ {1o, 2o}) |
5 | 1, 4 | mto 199 | . 2 ⊢ ¬ ∅ = 𝑋 |
6 | 5 | neir 3021 | 1 ⊢ ∅ ≠ 𝑋 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∅c0 4293 {cpr 4571 1oc1o 8097 2oc2o 8098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-nul 5212 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-v 3498 df-dif 3941 df-un 3943 df-nul 4294 df-sn 4570 df-pr 4572 df-suc 6199 df-1o 8104 df-2o 8105 |
This theorem is referenced by: sltres 33171 noextenddif 33177 nolesgn2ores 33181 nosepnelem 33186 nosepdmlem 33189 nolt02o 33201 nosupbnd1lem3 33212 nosupbnd1lem5 33214 nosupbnd2lem1 33217 |
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