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Mirrors > Home > MPE Home > Th. List > tsrps | Structured version Visualization version GIF version |
Description: A toset is a poset. (Contributed by Mario Carneiro, 9-Sep-2015.) |
Ref | Expression |
---|---|
tsrps | ⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . 3 ⊢ dom 𝑅 = dom 𝑅 | |
2 | 1 | istsr 18641 | . 2 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅 ∪ ◡𝑅))) |
3 | 2 | simplbi 497 | 1 ⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∪ cun 3961 ⊆ wss 3963 × cxp 5687 ◡ccnv 5688 dom cdm 5689 PosetRelcps 18622 TosetRel ctsr 18623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-tsr 18625 |
This theorem is referenced by: cnvtsr 18646 tsrdir 18662 ordtbas2 23215 ordtrest2lem 23227 ordtrest2 23228 ordthauslem 23407 icopnfhmeo 24988 iccpnfhmeo 24990 xrhmeo 24991 cnvordtrestixx 33874 xrge0iifhmeo 33897 |
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