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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 5582 | . 2 ⊢ ¬ ∅ ∈ (R × R) | |
2 | df-c 10531 | . . 3 ⊢ ℂ = (R × R) | |
3 | 2 | eleq2i 2901 | . 2 ⊢ (∅ ∈ ℂ ↔ ∅ ∈ (R × R)) |
4 | 1, 3 | mtbir 324 | 1 ⊢ ¬ ∅ ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2105 ∅c0 4288 × cxp 5546 Rcnr 10275 ℂcc 10523 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-opab 5120 df-xp 5554 df-c 10531 |
This theorem is referenced by: axaddf 10555 axmulf 10556 bj-inftyexpitaudisj 34379 bj-inftyexpidisj 34384 |
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