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| Mirrors > Home > MPE Home > Th. List > 0nrp | Structured version Visualization version GIF version | ||
| Description: Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.) |
| Ref | Expression |
|---|---|
| 0nrp | ⊢ ¬ 0 ∈ ℝ+ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0re 11106 | . . 3 ⊢ 0 ∈ ℝ | |
| 2 | 1 | ltnri 11214 | . 2 ⊢ ¬ 0 < 0 |
| 3 | rpgt0 12895 | . 2 ⊢ (0 ∈ ℝ+ → 0 < 0) | |
| 4 | 2, 3 | mto 197 | 1 ⊢ ¬ 0 ∈ ℝ+ |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∈ wcel 2110 class class class wbr 5089 0cc0 10998 < clt 11138 ℝ+crp 12882 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-resscn 11055 ax-1cn 11056 ax-addrcl 11059 ax-rnegex 11069 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-po 5522 df-so 5523 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-er 8617 df-en 8865 df-dom 8866 df-sdom 8867 df-pnf 11140 df-mnf 11141 df-ltxr 11143 df-rp 12883 |
| This theorem is referenced by: itg2gt0 25681 dvrelog 26566 logdmn0 26569 logdivsqrle 34653 dvrelog2 42076 dvrelog3 42077 elbigolo1 48568 |
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