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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 26958 | . . 3 ⊢ ¬ ∅ ∈ {1o, 2o} | |
2 | nosgnn0i.1 | . . . 4 ⊢ 𝑋 ∈ {1o, 2o} | |
3 | eleq1 2825 | . . . 4 ⊢ (∅ = 𝑋 → (∅ ∈ {1o, 2o} ↔ 𝑋 ∈ {1o, 2o})) | |
4 | 2, 3 | mpbiri 257 | . . 3 ⊢ (∅ = 𝑋 → ∅ ∈ {1o, 2o}) |
5 | 1, 4 | mto 196 | . 2 ⊢ ¬ ∅ = 𝑋 |
6 | 5 | neir 2944 | 1 ⊢ ∅ ≠ 𝑋 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ∅c0 4280 {cpr 4586 1oc1o 8397 2oc2o 8398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 ax-nul 5261 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2942 df-v 3445 df-dif 3911 df-un 3913 df-nul 4281 df-sn 4585 df-pr 4587 df-suc 6321 df-1o 8404 df-2o 8405 |
This theorem is referenced by: sltres 26962 noextenddif 26968 nolesgn2ores 26972 nosepnelem 26979 nosepdmlem 26983 nolt02o 26995 nosupbnd1lem3 27010 nosupbnd1lem5 27012 nosupbnd2lem1 27015 |
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