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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 2798 | . . 3 ⊢ dom 𝑅 = dom 𝑅 | |
2 | 1 | istsr 17819 | . 2 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅 ∪ ◡𝑅))) |
3 | 2 | simplbi 501 | 1 ⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ∪ cun 3879 ⊆ wss 3881 × cxp 5517 ◡ccnv 5518 dom cdm 5519 PosetRelcps 17800 TosetRel ctsr 17801 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-tsr 17803 |
This theorem is referenced by: cnvtsr 17824 tsrdir 17840 ordtbas2 21796 ordtrest2lem 21808 ordtrest2 21809 ordthauslem 21988 icopnfhmeo 23548 iccpnfhmeo 23550 xrhmeo 23551 cnvordtrestixx 31266 xrge0iifhmeo 31289 |
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