Theorem lejoin2 17891

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 17889	. 2 ⊢ (𝜑 → ((𝑋 ≤ (𝑋 ∨ 𝑌) ∧ 𝑌 ≤ (𝑋 ∨ 𝑌)) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧)))
9	8	simplrd 770	1 ⊢ (𝜑 → 𝑌 ≤ (𝑋 ∨ 𝑌))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ∀wral 3061  ⟨cop 4547   class class class wbr 5053  dom cdm 5551  ‘cfv 6380  (class class class)co 7213  Basecbs 16760  lecple 16809  joincjn 17818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-lub 17852  df-join 17854

This theorem is referenced by:  joinle  17892  latlej2  17955