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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 8525 | . . . 4 ⊢ 1o ≠ ∅ | |
2 | 1 | nesymi 2996 | . . 3 ⊢ ¬ ∅ = 1o |
3 | nsuceq0 6469 | . . . . 5 ⊢ suc 1o ≠ ∅ | |
4 | necom 2992 | . . . . . 6 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o) | |
5 | df-2o 8506 | . . . . . . 7 ⊢ 2o = suc 1o | |
6 | 5 | neeq2i 3004 | . . . . . 6 ⊢ (∅ ≠ 2o ↔ ∅ ≠ suc 1o) |
7 | 4, 6 | bitr4i 278 | . . . . 5 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ 2o) |
8 | 3, 7 | mpbi 230 | . . . 4 ⊢ ∅ ≠ 2o |
9 | 8 | neii 2940 | . . 3 ⊢ ¬ ∅ = 2o |
10 | 2, 9 | pm3.2ni 880 | . 2 ⊢ ¬ (∅ = 1o ∨ ∅ = 2o) |
11 | 0ex 5313 | . . 3 ⊢ ∅ ∈ V | |
12 | 11 | elpr 4655 | . 2 ⊢ (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o)) |
13 | 10, 12 | mtbir 323 | 1 ⊢ ¬ ∅ ∈ {1o, 2o} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 {cpr 4633 suc csuc 6388 1oc1o 8498 2oc2o 8499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-v 3480 df-dif 3966 df-un 3968 df-nul 4340 df-sn 4632 df-pr 4634 df-suc 6392 df-1o 8505 df-2o 8506 |
This theorem is referenced by: nosgnn0i 27719 sltres 27722 noseponlem 27724 sltso 27736 nosepssdm 27746 nodenselem8 27751 nolt02olem 27754 |
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