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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 5672 | . 2 ⊢ ¬ ∅ ∈ (R × R) | |
2 | df-c 11064 | . . 3 ⊢ ℂ = (R × R) | |
3 | 2 | eleq2i 2830 | . 2 ⊢ (∅ ∈ ℂ ↔ ∅ ∈ (R × R)) |
4 | 1, 3 | mtbir 323 | 1 ⊢ ¬ ∅ ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2107 ∅c0 4287 × cxp 5636 Rcnr 10808 ℂcc 11056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-opab 5173 df-xp 5644 df-c 11064 |
This theorem is referenced by: axaddf 11088 axmulf 11089 bj-inftyexpitaudisj 35705 bj-inftyexpidisj 35710 |
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