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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 2821 | . . 3 ⊢ dom 𝑅 = dom 𝑅 | |
2 | 1 | istsr 17821 | . 2 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅 ∪ ◡𝑅))) |
3 | 2 | simplbi 500 | 1 ⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ∪ cun 3934 ⊆ wss 3936 × cxp 5548 ◡ccnv 5549 dom cdm 5550 PosetRelcps 17802 TosetRel ctsr 17803 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-xp 5556 df-cnv 5558 df-dm 5560 df-tsr 17805 |
This theorem is referenced by: cnvtsr 17826 tsrdir 17842 ordtbas2 21793 ordtrest2lem 21805 ordtrest2 21806 ordthauslem 21985 icopnfhmeo 23541 iccpnfhmeo 23543 xrhmeo 23544 cnvordtrestixx 31151 xrge0iifhmeo 31174 |
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