Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2622 | . . 3 ⊢ dom 𝑅 = dom 𝑅 | |
2 | 1 | istsr 17217 | . 2 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅 ∪ ◡𝑅))) |
3 | 2 | simplbi 476 | 1 ⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ∪ cun 3572 ⊆ wss 3574 × cxp 5112 ◡ccnv 5113 dom cdm 5114 PosetRelcps 17198 TosetRel ctsr 17199 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-tsr 17201 |
This theorem is referenced by: cnvtsr 17222 tsrdir 17238 ordtbas2 20995 ordtrest2lem 21007 ordtrest2 21008 ordthauslem 21187 icopnfhmeo 22742 iccpnfhmeo 22744 xrhmeo 22745 cnvordtrestixx 29959 xrge0iifhmeo 29982 |
Copyright terms: Public domain | W3C validator |