Theorem lejoin2 18348

Description: A join's second argument is less than or equal to the join. (Contributed by NM, 16-Sep-2011.)

Hypotheses

Ref	Expression
joinval2.b	⊢ 𝐵 = (Base‘𝐾)
joinval2.l	⊢ ≤ = (le‘𝐾)
joinval2.j	⊢ ∨ = (join‘𝐾)
joinval2.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
joinval2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
joinval2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
joinlem.e	⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∨ )

Assertion

Ref	Expression
lejoin2	⊢ (𝜑 → 𝑌 ≤ (𝑋 ∨ 𝑌))

Proof of Theorem lejoin2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		joinval2.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		joinval2.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		joinval2.j	. . 3 ⊢ ∨ = (join‘𝐾)
4		joinval2.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
5		joinval2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		joinval2.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7		joinlem.e	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∨ )
8	1, 2, 3, 4, 5, 6, 7	joinlem 18346	. 2 ⊢ (𝜑 → ((𝑋 ≤ (𝑋 ∨ 𝑌) ∧ 𝑌 ≤ (𝑋 ∨ 𝑌)) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧)))
9	8	simplrd 767	1 ⊢ (𝜑 → 𝑌 ≤ (𝑋 ∨ 𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060  ⟨cop 4634   class class class wbr 5148  dom cdm 5676  ‘cfv 6543  (class class class)co 7412  Basecbs 17151  lecple 17211  joincjn 18274

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-lub 18309  df-join 18311

This theorem is referenced by:  joinle  18349  latlej2  18412