Theorem prsref 17544

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 1124	. . 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 17543	. . 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 497	1 ⊢ ((𝐾 ∈ Proset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   class class class wbr 5068  ‘cfv 6357  Basecbs 16485  lecple 16574   Proset cproset 17538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-nul 5212

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365  df-proset 17540

This theorem is referenced by:  posref  17563  prsdm  31159  prsrn  31160