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Mirrors > Home > MPE Home > Th. List > ex-po | Structured version Visualization version GIF version |
Description: Example for df-po 5233. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.) |
Ref | Expression |
---|---|
ex-po | ⊢ ( < Po ℝ ∧ ¬ ≤ Po ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltso 10408 | . . 3 ⊢ < Or ℝ | |
2 | sopo 5250 | . . 3 ⊢ ( < Or ℝ → < Po ℝ) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ < Po ℝ |
4 | 0le0 11421 | . . 3 ⊢ 0 ≤ 0 | |
5 | 0re 10330 | . . . 4 ⊢ 0 ∈ ℝ | |
6 | poirr 5244 | . . . 4 ⊢ (( ≤ Po ℝ ∧ 0 ∈ ℝ) → ¬ 0 ≤ 0) | |
7 | 5, 6 | mpan2 683 | . . 3 ⊢ ( ≤ Po ℝ → ¬ 0 ≤ 0) |
8 | 4, 7 | mt2 192 | . 2 ⊢ ¬ ≤ Po ℝ |
9 | 3, 8 | pm3.2i 463 | 1 ⊢ ( < Po ℝ ∧ ¬ ≤ Po ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 385 ∈ wcel 2157 class class class wbr 4843 Po wpo 5231 Or wor 5232 ℝcr 10223 0cc0 10224 < clt 10363 ≤ cle 10364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-resscn 10281 ax-1cn 10282 ax-addrcl 10285 ax-rnegex 10295 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-po 5233 df-so 5234 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 |
This theorem is referenced by: (None) |
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