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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 8502 | . . . 4 ⊢ 1o ≠ ∅ | |
2 | 1 | nesymi 2993 | . . 3 ⊢ ¬ ∅ = 1o |
3 | nsuceq0 6446 | . . . . 5 ⊢ suc 1o ≠ ∅ | |
4 | necom 2989 | . . . . . 6 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o) | |
5 | df-2o 8481 | . . . . . . 7 ⊢ 2o = suc 1o | |
6 | 5 | neeq2i 3001 | . . . . . 6 ⊢ (∅ ≠ 2o ↔ ∅ ≠ suc 1o) |
7 | 4, 6 | bitr4i 278 | . . . . 5 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ 2o) |
8 | 3, 7 | mpbi 229 | . . . 4 ⊢ ∅ ≠ 2o |
9 | 8 | neii 2937 | . . 3 ⊢ ¬ ∅ = 2o |
10 | 2, 9 | pm3.2ni 879 | . 2 ⊢ ¬ (∅ = 1o ∨ ∅ = 2o) |
11 | 0ex 5301 | . . 3 ⊢ ∅ ∈ V | |
12 | 11 | elpr 4647 | . 2 ⊢ (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o)) |
13 | 10, 12 | mtbir 323 | 1 ⊢ ¬ ∅ ∈ {1o, 2o} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∅c0 4318 {cpr 4626 suc csuc 6365 1oc1o 8473 2oc2o 8474 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-nul 5300 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2936 df-v 3471 df-dif 3947 df-un 3949 df-nul 4319 df-sn 4625 df-pr 4627 df-suc 6369 df-1o 8480 df-2o 8481 |
This theorem is referenced by: nosgnn0i 27579 sltres 27582 noseponlem 27584 sltso 27596 nosepssdm 27606 nodenselem8 27611 nolt02olem 27614 |
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