Theorem lejoin2 17452

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 17450	. 2 ⊢ (𝜑 → ((𝑋 ≤ (𝑋 ∨ 𝑌) ∧ 𝑌 ≤ (𝑋 ∨ 𝑌)) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧)))
9	8	simplrd 766	1 ⊢ (𝜑 → 𝑌 ≤ (𝑋 ∨ 𝑌))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1522   ∈ wcel 2081  ∀wral 3105  ⟨cop 4478   class class class wbr 4962  dom cdm 5443  ‘cfv 6225  (class class class)co 7016  Basecbs 16312  lecple 16401  joincjn 17383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-lub 17413  df-join 17415

This theorem is referenced by:  joinle  17453  latlej2  17500