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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 18374 | . 2 ⊢ (𝐾 ∈ Poset → 𝐾 ∈ Proset ) | |
2 | posi.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
3 | posi.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
4 | 2, 3 | prsref 18356 | . 2 ⊢ ((𝐾 ∈ Proset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋) |
5 | 1, 4 | sylan 580 | 1 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 Basecbs 17245 lecple 17305 Proset cproset 18350 Posetcpo 18365 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-proset 18352 df-poset 18371 |
This theorem is referenced by: posasymb 18377 odupos 18386 pleval2 18395 pltval3 18397 pospo 18403 lublecllem 18418 latref 18499 omndmul2 33072 omndmul 33074 gsumle 33084 archirngz 33179 cvrnbtwn2 39257 cvrnbtwn3 39258 cvrnbtwn4 39261 cvrcmp 39265 llncmp 39505 lplncmp 39545 lvolcmp 39600 lubprlem 48759 posjidm 48769 posmidm 48770 |
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