![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > posref | Structured version Visualization version GIF version |
Description: A poset ordering is reflexive. (Contributed by NM, 11-Sep-2011.) (Proof shortened by OpenAI, 25-Mar-2020.) |
Ref | Expression |
---|---|
posi.b | ⊢ 𝐵 = (Base‘𝐾) |
posi.l | ⊢ ≤ = (le‘𝐾) |
Ref | Expression |
---|---|
posref | ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | posprs 17551 | . 2 ⊢ (𝐾 ∈ Poset → 𝐾 ∈ Proset ) | |
2 | posi.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
3 | posi.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
4 | 2, 3 | prsref 17534 | . 2 ⊢ ((𝐾 ∈ Proset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋) |
5 | 1, 4 | sylan 583 | 1 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 Basecbs 16475 lecple 16564 Proset cproset 17528 Posetcpo 17542 |
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 ax-nul 5174 |
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-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-proset 17530 df-poset 17548 |
This theorem is referenced by: posasymb 17554 pleval2 17567 pltval3 17569 pospo 17575 lublecllem 17590 latref 17655 odupos 17737 omndmul2 30763 omndmul 30765 gsumle 30775 archirngz 30868 cvrnbtwn2 36571 cvrnbtwn3 36572 cvrnbtwn4 36575 cvrcmp 36579 llncmp 36818 lplncmp 36858 lvolcmp 36913 |
Copyright terms: Public domain | W3C validator |