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| Mirrors > Home > MPE Home > Th. List > nosgnn0 | Structured version Visualization version GIF version | ||
| Description: ∅ is not a surreal sign. (Contributed by Scott Fenton, 16-Jun-2011.) | 
| Ref | Expression | 
|---|---|
| nosgnn0 | ⊢ ¬ ∅ ∈ {1o, 2o} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 1n0 8526 | . . . 4 ⊢ 1o ≠ ∅ | |
| 2 | 1 | nesymi 2998 | . . 3 ⊢ ¬ ∅ = 1o | 
| 3 | nsuceq0 6467 | . . . . 5 ⊢ suc 1o ≠ ∅ | |
| 4 | necom 2994 | . . . . . 6 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o) | |
| 5 | df-2o 8507 | . . . . . . 7 ⊢ 2o = suc 1o | |
| 6 | 5 | neeq2i 3006 | . . . . . 6 ⊢ (∅ ≠ 2o ↔ ∅ ≠ suc 1o) | 
| 7 | 4, 6 | bitr4i 278 | . . . . 5 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ 2o) | 
| 8 | 3, 7 | mpbi 230 | . . . 4 ⊢ ∅ ≠ 2o | 
| 9 | 8 | neii 2942 | . . 3 ⊢ ¬ ∅ = 2o | 
| 10 | 2, 9 | pm3.2ni 881 | . 2 ⊢ ¬ (∅ = 1o ∨ ∅ = 2o) | 
| 11 | 0ex 5307 | . . 3 ⊢ ∅ ∈ V | |
| 12 | 11 | elpr 4650 | . 2 ⊢ (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o)) | 
| 13 | 10, 12 | mtbir 323 | 1 ⊢ ¬ ∅ ∈ {1o, 2o} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 {cpr 4628 suc csuc 6386 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: nosgnn0i 27704 sltres 27707 noseponlem 27709 sltso 27721 nosepssdm 27731 nodenselem8 27736 nolt02olem 27739 | 
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