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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 5389 | . 2 ⊢ ¬ ∅ ∈ (R × R) | |
2 | df-c 10278 | . . 3 ⊢ ℂ = (R × R) | |
3 | 2 | eleq2i 2850 | . 2 ⊢ (∅ ∈ ℂ ↔ ∅ ∈ (R × R)) |
4 | 1, 3 | mtbir 315 | 1 ⊢ ¬ ∅ ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2106 ∅c0 4140 × cxp 5353 Rcnr 10022 ℂcc 10270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-v 3399 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-opab 4949 df-xp 5361 df-c 10278 |
This theorem is referenced by: axaddf 10302 axmulf 10303 bj-inftyexpitaudisj 33696 bj-inftyexpidisj 33701 |
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