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Mirrors > Home > MPE Home > Th. List > ex-po | Structured version Visualization version GIF version |
Description: Example for df-po 5586. 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 11335 | . . 3 ⊢ < Or ℝ | |
2 | sopo 5605 | . . 3 ⊢ ( < Or ℝ → < Po ℝ) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ < Po ℝ |
4 | 0le0 12359 | . . 3 ⊢ 0 ≤ 0 | |
5 | 0re 11257 | . . . 4 ⊢ 0 ∈ ℝ | |
6 | poirr 5598 | . . . 4 ⊢ (( ≤ Po ℝ ∧ 0 ∈ ℝ) → ¬ 0 ≤ 0) | |
7 | 5, 6 | mpan2 689 | . . 3 ⊢ ( ≤ Po ℝ → ¬ 0 ≤ 0) |
8 | 4, 7 | mt2 199 | . 2 ⊢ ¬ ≤ Po ℝ |
9 | 3, 8 | pm3.2i 469 | 1 ⊢ ( < Po ℝ ∧ ¬ ≤ Po ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 394 ∈ wcel 2099 class class class wbr 5145 Po wpo 5584 Or wor 5585 ℝcr 11148 0cc0 11149 < clt 11289 ≤ cle 11290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-resscn 11206 ax-1cn 11207 ax-addrcl 11210 ax-rnegex 11220 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 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 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 |
This theorem is referenced by: (None) |
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