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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 8318 | . . . 4 ⊢ 1o ≠ ∅ | |
2 | 1 | nesymi 3001 | . . 3 ⊢ ¬ ∅ = 1o |
3 | nsuceq0 6346 | . . . . 5 ⊢ suc 1o ≠ ∅ | |
4 | necom 2997 | . . . . . 6 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o) | |
5 | df-2o 8298 | . . . . . . 7 ⊢ 2o = suc 1o | |
6 | 5 | neeq2i 3009 | . . . . . 6 ⊢ (∅ ≠ 2o ↔ ∅ ≠ suc 1o) |
7 | 4, 6 | bitr4i 277 | . . . . 5 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ 2o) |
8 | 3, 7 | mpbi 229 | . . . 4 ⊢ ∅ ≠ 2o |
9 | 8 | neii 2945 | . . 3 ⊢ ¬ ∅ = 2o |
10 | 2, 9 | pm3.2ni 878 | . 2 ⊢ ¬ (∅ = 1o ∨ ∅ = 2o) |
11 | 0ex 5231 | . . 3 ⊢ ∅ ∈ V | |
12 | 11 | elpr 4584 | . 2 ⊢ (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o)) |
13 | 10, 12 | mtbir 323 | 1 ⊢ ¬ ∅ ∈ {1o, 2o} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 844 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∅c0 4256 {cpr 4563 suc csuc 6268 1oc1o 8290 2oc2o 8291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-nul 5230 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-v 3434 df-dif 3890 df-un 3892 df-nul 4257 df-sn 4562 df-pr 4564 df-suc 6272 df-1o 8297 df-2o 8298 |
This theorem is referenced by: nosgnn0i 33862 sltres 33865 noseponlem 33867 sltso 33879 nosepssdm 33889 nodenselem8 33894 nolt02olem 33897 |
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