Theorem prsref 18345

Description: "Less than or equal to" is reflexive in a proset. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Hypotheses

Ref	Expression
isprs.b	⊢ 𝐵 = (Base‘𝐾)
isprs.l	⊢ ≤ = (le‘𝐾)

Assertion

Ref	Expression
prsref	⊢ ((𝐾 ∈ Proset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)

Proof of Theorem prsref

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)

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