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| Description: "Less than or equal to" is reflexive in a proset. (Contributed by Stefan O'Rear, 1-Feb-2015.) | 
| Ref | Expression | 
|---|---|
| isprs.b | ⊢ 𝐵 = (Base‘𝐾) | 
| isprs.l | ⊢ ≤ = (le‘𝐾) | 
| Ref | Expression | 
|---|---|
| prsref | ⊢ ((𝐾 ∈ Proset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | id 22 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐵) | |
| 2 | 1, 1, 1 | 3jca 1128 | . . 3 ⊢ (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) | 
| 3 | isprs.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
| 4 | isprs.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
| 5 | 3, 4 | prslem 18344 | . . 3 ⊢ ((𝐾 ∈ Proset ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑋 ∧ 𝑋 ≤ 𝑋) → 𝑋 ≤ 𝑋))) | 
| 6 | 2, 5 | sylan2 593 | . 2 ⊢ ((𝐾 ∈ Proset ∧ 𝑋 ∈ 𝐵) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑋 ∧ 𝑋 ≤ 𝑋) → 𝑋 ≤ 𝑋))) | 
| 7 | 6 | simpld 494 | 1 ⊢ ((𝐾 ∈ Proset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 class class class wbr 5142 ‘cfv 6560 Basecbs 17248 lecple 17305 Proset cproset 18339 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-nul 5305 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-proset 18341 | 
| This theorem is referenced by: posref 18365 mgccole1 32981 mgccole2 32982 prsdm 33914 prsrn 33915 prsthinc 49136 | 
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