Theorem postr 18377

Description: A poset ordering is transitive. (Contributed by NM, 11-Sep-2011.)

Hypotheses

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

Assertion

Ref	Expression
postr	⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍))

Proof of Theorem postr

Step	Hyp	Ref	Expression
1		posi.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		posi.l	. . 3 ⊢ ≤ = (le‘𝐾)
3	1, 2	posi 18374	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌) ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)))
4	3	simp3d 1162	1 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   class class class wbr 5110  ‘cfv 6538  Basecbs 17270  lecple 17318  Posetcpo 18364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5270

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-iota 6494  df-fv 6546  df-poset 18370

This theorem is referenced by:  odupos  18383  plttr  18397  joinle  18441  meetle  18455  lattr  18501  omndadd2d  20201  omndadd2rd  20202  omndmul2  20204  atlatle  40075  cvratlem  40176  llncmp  40277  llncvrlpln  40313  lplncmp  40317  lplncvrlvol  40371  lvolcmp  40372  pmaple  40516