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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 31055 | . . 3 ⊢ ¬ ∅ ∈ {1𝑜, 2𝑜} | |
2 | nosgnn0i.1 | . . . 4 ⊢ 𝑋 ∈ {1𝑜, 2𝑜} | |
3 | eleq1 2676 | . . . 4 ⊢ (∅ = 𝑋 → (∅ ∈ {1𝑜, 2𝑜} ↔ 𝑋 ∈ {1𝑜, 2𝑜})) | |
4 | 2, 3 | mpbiri 247 | . . 3 ⊢ (∅ = 𝑋 → ∅ ∈ {1𝑜, 2𝑜}) |
5 | 1, 4 | mto 187 | . 2 ⊢ ¬ ∅ = 𝑋 |
6 | 5 | neir 2785 | 1 ⊢ ∅ ≠ 𝑋 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 {cpr 4127 1𝑜c1o 7440 2𝑜c2o 7441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-un 3545 df-nul 3875 df-sn 4126 df-pr 4128 df-suc 5646 df-1o 7447 df-2o 7448 |
This theorem is referenced by: sltres 31061 nobndlem2 31092 nobndlem4 31094 nobndlem5 31095 nobndlem6 31096 nobndlem8 31098 |
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