Theorem lemeet2 18098

Description: A meet's second argument is less than or equal to the meet. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 12-Sep-2018.)

Hypotheses

Ref	Expression
meetval2.b	⊢ 𝐵 = (Base‘𝐾)
meetval2.l	⊢ ≤ = (le‘𝐾)
meetval2.m	⊢ ∧ = (meet‘𝐾)
meetval2.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
meetval2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
meetval2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
meetlem.e	⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∧ )

Assertion

Ref	Expression
lemeet2	⊢ (𝜑 → (𝑋 ∧ 𝑌) ≤ 𝑌)

Proof of Theorem lemeet2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		meetval2.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		meetval2.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		meetval2.m	. . 3 ⊢ ∧ = (meet‘𝐾)
4		meetval2.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
5		meetval2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		meetval2.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7		meetlem.e	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∧ )
8	1, 2, 3, 4, 5, 6, 7	meetlem 18096	. 2 ⊢ (𝜑 → (((𝑋 ∧ 𝑌) ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌))))
9	8	simplrd 766	1 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ≤ 𝑌)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2109  ∀wral 3065  ⟨cop 4572   class class class wbr 5078  dom cdm 5588  ‘cfv 6430  (class class class)co 7268  Basecbs 16893  lecple 16950  meetcmee 18011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-riota 7225  df-ov 7271  df-oprab 7272  df-glb 18046  df-meet 18048

This theorem is referenced by:  meetle  18099  latmle2  18164