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Theorem lemeet2 18362
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 𝐡 = (Baseβ€˜πΎ)
meetval2.l ≀ = (leβ€˜πΎ)
meetval2.m ∧ = (meetβ€˜πΎ)
meetval2.k (πœ‘ β†’ 𝐾 ∈ 𝑉)
meetval2.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
meetval2.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
meetlem.e (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∧ )
Assertion
Ref Expression
lemeet2 (πœ‘ β†’ (𝑋 ∧ π‘Œ) ≀ π‘Œ)

Proof of Theorem lemeet2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 meetval2.b . . 3 𝐡 = (Baseβ€˜πΎ)
2 meetval2.l . . 3 ≀ = (leβ€˜πΎ)
3 meetval2.m . . 3 ∧ = (meetβ€˜πΎ)
4 meetval2.k . . 3 (πœ‘ β†’ 𝐾 ∈ 𝑉)
5 meetval2.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐡)
6 meetval2.y . . 3 (πœ‘ β†’ π‘Œ ∈ 𝐡)
7 meetlem.e . . 3 (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∧ )
81, 2, 3, 4, 5, 6, 7meetlem 18360 . 2 (πœ‘ β†’ (((𝑋 ∧ π‘Œ) ≀ 𝑋 ∧ (𝑋 ∧ π‘Œ) ≀ π‘Œ) ∧ βˆ€π‘§ ∈ 𝐡 ((𝑧 ≀ 𝑋 ∧ 𝑧 ≀ π‘Œ) β†’ 𝑧 ≀ (𝑋 ∧ π‘Œ))))
98simplrd 767 1 (πœ‘ β†’ (𝑋 ∧ π‘Œ) ≀ π‘Œ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  βŸ¨cop 4629   class class class wbr 5141  dom cdm 5669  β€˜cfv 6536  (class class class)co 7404  Basecbs 17151  lecple 17211  meetcmee 18275
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 18310  df-meet 18312
This theorem is referenced by:  meetle  18363  latmle2  18428
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