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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 5723 | . 2 ⊢ ¬ ∅ ∈ (R × R) | |
2 | df-c 11159 | . . 3 ⊢ ℂ = (R × R) | |
3 | 2 | eleq2i 2831 | . 2 ⊢ (∅ ∈ ℂ ↔ ∅ ∈ (R × R)) |
4 | 1, 3 | mtbir 323 | 1 ⊢ ¬ ∅ ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2106 ∅c0 4339 × cxp 5687 Rcnr 10903 ℂcc 11151 |
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-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 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-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-opab 5211 df-xp 5695 df-c 11159 |
This theorem is referenced by: axaddf 11183 axmulf 11184 bj-inftyexpitaudisj 37188 bj-inftyexpidisj 37193 |
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