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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 2738 | . . 3 ⊢ dom 𝑅 = dom 𝑅 | |
2 | 1 | istsr 17936 | . 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 2113 ∪ cun 3839 ⊆ wss 3841 × cxp 5517 ◡ccnv 5518 dom cdm 5519 PosetRelcps 17917 TosetRel ctsr 17918 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-rab 3062 df-v 3399 df-un 3846 df-in 3848 df-ss 3858 df-sn 4514 df-pr 4516 df-op 4520 df-br 5028 df-opab 5090 df-xp 5525 df-cnv 5527 df-dm 5529 df-tsr 17920 |
This theorem is referenced by: cnvtsr 17941 tsrdir 17957 ordtbas2 21935 ordtrest2lem 21947 ordtrest2 21948 ordthauslem 22127 icopnfhmeo 23688 iccpnfhmeo 23690 xrhmeo 23691 cnvordtrestixx 31427 xrge0iifhmeo 31450 |
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