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Mirrors > Home > MPE Home > Th. List > gtso | Structured version Visualization version GIF version |
Description: 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.) |
Ref | Expression |
---|---|
gtso | ⊢ ◡ < Or ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltso 11335 | . 2 ⊢ < Or ℝ | |
2 | cnvso 6291 | . 2 ⊢ ( < Or ℝ ↔ ◡ < Or ℝ) | |
3 | 1, 2 | mpbi 229 | 1 ⊢ ◡ < Or ℝ |
Colors of variables: wff setvar class |
Syntax hints: Or wor 5585 ◡ccnv 5673 ℝcr 11148 < clt 11289 |
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-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-ltxr 11294 |
This theorem is referenced by: infrenegsup 12243 dvlt0 26026 erdszelem9 35040 erdszelem11 35042 rencldnfilem 42514 |
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