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| Mirrors > Home > MPE Home > Th. List > 0ncn | Structured version Visualization version GIF version | ||
| Description: The empty set is not a complex number. Note: do not use this after the real number axioms are developed, since it is a construction-dependent property. (Contributed by NM, 2-May-1996.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| 0ncn | ⊢ ¬ ∅ ∈ ℂ | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 0nelxp 5718 | . 2 ⊢ ¬ ∅ ∈ (R × R) | |
| 2 | df-c 11162 | . . 3 ⊢ ℂ = (R × R) | |
| 3 | 2 | eleq2i 2832 | . 2 ⊢ (∅ ∈ ℂ ↔ ∅ ∈ (R × R)) | 
| 4 | 1, 3 | mtbir 323 | 1 ⊢ ¬ ∅ ∈ ℂ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∈ wcel 2107 ∅c0 4332 × cxp 5682 Rcnr 10906 ℂcc 11154 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-opab 5205 df-xp 5690 df-c 11162 | 
| This theorem is referenced by: axaddf 11186 axmulf 11187 bj-inftyexpitaudisj 37207 bj-inftyexpidisj 37212 | 
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