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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 8111 | . . . 4 ⊢ 1o ≠ ∅ | |
2 | 1 | nesymi 3071 | . . 3 ⊢ ¬ ∅ = 1o |
3 | nsuceq0 6264 | . . . . 5 ⊢ suc 1o ≠ ∅ | |
4 | necom 3067 | . . . . . 6 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o) | |
5 | df-2o 8095 | . . . . . . 7 ⊢ 2o = suc 1o | |
6 | 5 | neeq2i 3079 | . . . . . 6 ⊢ (∅ ≠ 2o ↔ ∅ ≠ suc 1o) |
7 | 4, 6 | bitr4i 280 | . . . . 5 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ 2o) |
8 | 3, 7 | mpbi 232 | . . . 4 ⊢ ∅ ≠ 2o |
9 | 8 | neii 3016 | . . 3 ⊢ ¬ ∅ = 2o |
10 | 2, 9 | pm3.2ni 877 | . 2 ⊢ ¬ (∅ = 1o ∨ ∅ = 2o) |
11 | 0ex 5202 | . . 3 ⊢ ∅ ∈ V | |
12 | 11 | elpr 4582 | . 2 ⊢ (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o)) |
13 | 10, 12 | mtbir 325 | 1 ⊢ ¬ ∅ ∈ {1o, 2o} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 843 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 ∅c0 4289 {cpr 4561 suc csuc 6186 1oc1o 8087 2oc2o 8088 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-nul 5201 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-v 3495 df-dif 3937 df-un 3939 df-nul 4290 df-sn 4560 df-pr 4562 df-suc 6190 df-1o 8094 df-2o 8095 |
This theorem is referenced by: nosgnn0i 33159 sltres 33162 noseponlem 33164 sltso 33174 nosepssdm 33183 nodenselem8 33188 nolt02olem 33191 |
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