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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 5701 | . 2 ⊢ ¬ ∅ ∈ (R × R) | |
2 | df-c 11113 | . . 3 ⊢ ℂ = (R × R) | |
3 | 2 | eleq2i 2817 | . 2 ⊢ (∅ ∈ ℂ ↔ ∅ ∈ (R × R)) |
4 | 1, 3 | mtbir 323 | 1 ⊢ ¬ ∅ ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2098 ∅c0 4315 × cxp 5665 Rcnr 10857 ℂcc 11105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-opab 5202 df-xp 5673 df-c 11113 |
This theorem is referenced by: axaddf 11137 axmulf 11138 bj-inftyexpitaudisj 36577 bj-inftyexpidisj 36582 |
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