Theorem lemeet1 18397

Description: A meet's first 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
lemeet1	⊢ (𝜑 → (𝑋 ∧ 𝑌) ≤ 𝑋)

Proof of Theorem lemeet1

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 18396	. 2 ⊢ (𝜑 → (((𝑋 ∧ 𝑌) ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌))))
9	8	simplld 766	1 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ≤ 𝑋)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3058  ⟨cop 4638   class class class wbr 5152  dom cdm 5682  ‘cfv 6553  (class class class)co 7426  Basecbs 17187  lecple 17247  meetcmee 18311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-glb 18346  df-meet 18348

This theorem is referenced by:  meetle  18399  latmle1  18463