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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 17554 | . 2 ⊢ (𝐾 ∈ Poset → 𝐾 ∈ Proset ) | |
2 | posi.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
3 | posi.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
4 | 2, 3 | prsref 17537 | . 2 ⊢ ((𝐾 ∈ Proset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋) |
5 | 1, 4 | sylan 582 | 1 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 class class class wbr 5059 ‘cfv 6348 Basecbs 16478 lecple 16567 Proset cproset 17531 Posetcpo 17545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-nul 5203 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-iota 6307 df-fv 6356 df-proset 17533 df-poset 17551 |
This theorem is referenced by: posasymb 17557 pleval2 17570 pltval3 17572 pospo 17578 lublecllem 17593 latref 17658 odupos 17740 omndmul2 30734 omndmul 30736 gsumle 30746 archirngz 30839 cvrnbtwn2 36444 cvrnbtwn3 36445 cvrnbtwn4 36448 cvrcmp 36452 llncmp 36691 lplncmp 36731 lvolcmp 36786 |
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