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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 7847 | . . . 4 ⊢ 1o ≠ ∅ | |
2 | 1 | nesymi 3056 | . . 3 ⊢ ¬ ∅ = 1o |
3 | nsuceq0 6047 | . . . . 5 ⊢ suc 1o ≠ ∅ | |
4 | necom 3052 | . . . . . 6 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o) | |
5 | df-2o 7832 | . . . . . . 7 ⊢ 2o = suc 1o | |
6 | 5 | neeq2i 3064 | . . . . . 6 ⊢ (∅ ≠ 2o ↔ ∅ ≠ suc 1o) |
7 | 4, 6 | bitr4i 270 | . . . . 5 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ 2o) |
8 | 3, 7 | mpbi 222 | . . . 4 ⊢ ∅ ≠ 2o |
9 | 8 | neii 3001 | . . 3 ⊢ ¬ ∅ = 2o |
10 | 2, 9 | pm3.2ni 909 | . 2 ⊢ ¬ (∅ = 1o ∨ ∅ = 2o) |
11 | 0ex 5016 | . . 3 ⊢ ∅ ∈ V | |
12 | 11 | elpr 4422 | . 2 ⊢ (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o)) |
13 | 10, 12 | mtbir 315 | 1 ⊢ ¬ ∅ ∈ {1o, 2o} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 878 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 ∅c0 4146 {cpr 4401 suc csuc 5969 1oc1o 7824 2oc2o 7825 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-nul 5015 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-v 3416 df-dif 3801 df-un 3803 df-nul 4147 df-sn 4400 df-pr 4402 df-suc 5973 df-1o 7831 df-2o 7832 |
This theorem is referenced by: nosgnn0i 32346 sltres 32349 noseponlem 32351 sltso 32361 nosepssdm 32370 nodenselem8 32375 nolt02olem 32378 |
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