Theorem prsref 18264

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 1129	. . 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 18263	. . 3 ⊢ ((𝐾 ∈ Proset ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑋 ∧ 𝑋 ≤ 𝑋) → 𝑋 ≤ 𝑋)))
6	2, 5	sylan2 594	. 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 1087   = wceq 1542   ∈ wcel 2114   class class class wbr 5085  ‘cfv 6498  Basecbs 17179  lecple 17227   Proset cproset 18258

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-proset 18260

This theorem is referenced by:  posref  18284  mgccole1  33050  mgccole2  33051  prsdm  34058  prsrn  34059  prsthinc  49939