Theorem lemeet1 18360

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 18359	. 2 ⊢ (𝜑 → (((𝑋 ∧ 𝑌) ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌))))
9	8	simplld 765	1 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ≤ 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3055  ⟨cop 4629   class class class wbr 5141  dom cdm 5669  ‘cfv 6536  (class class class)co 7404  Basecbs 17150  lecple 17210  meetcmee 18274

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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-glb 18309  df-meet 18311

This theorem is referenced by:  meetle  18362  latmle1  18426