Theorem lemeet1 18319

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 18318	. 2 ⊢ (𝜑 → (((𝑋 ∧ 𝑌) ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌))))
9	8	simplld 767	1 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ≤ 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ⟨cop 4586   class class class wbr 5098  dom cdm 5624  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  lecple 17184  meetcmee 18235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-glb 18268  df-meet 18270

This theorem is referenced by:  meetle  18321  latmle1  18387