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Mirrors > Home > MPE Home > Th. List > rpcndif0 | Structured version Visualization version GIF version |
Description: A positive real number is a complex number not being 0. (Contributed by AV, 29-May-2020.) |
Ref | Expression |
---|---|
rpcndif0 | ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ (ℂ ∖ {0})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpcnne0 12987 | . 2 ⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) | |
2 | eldifsn 4788 | . 2 ⊢ (𝐴 ∈ (ℂ ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) | |
3 | 1, 2 | sylibr 233 | 1 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ (ℂ ∖ {0})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 ≠ wne 2941 ∖ cdif 3943 {csn 4626 ℂcc 11103 0cc0 11105 ℝ+crp 12969 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-resscn 11162 ax-1cn 11163 ax-addrcl 11166 ax-rnegex 11176 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-po 5586 df-so 5587 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11245 df-mnf 11246 df-ltxr 11248 df-rp 12970 |
This theorem is referenced by: reefgim 25943 relogbreexp 26259 relogbmul 26261 relogbdiv 26263 relogbcxpb 26271 relogbf 26275 logbgt0b 26277 amgmlemALT 47751 |
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