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Theorem cvrletrN 38138
Description: Property of an element above a covering. (Contributed by NM, 7-Dec-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cvrletr.b 𝐡 = (Baseβ€˜πΎ)
cvrletr.l ≀ = (leβ€˜πΎ)
cvrletr.s < = (ltβ€˜πΎ)
cvrletr.c 𝐢 = ( β‹– β€˜πΎ)
Assertion
Ref Expression
cvrletrN ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((π‘‹πΆπ‘Œ ∧ π‘Œ ≀ 𝑍) β†’ 𝑋 < 𝑍))
Proof of Theorem cvrletrN
StepHypRef Expression
1 simpll 765 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ π‘‹πΆπ‘Œ) β†’ 𝐾 ∈ Poset)
2 simplr1 1215 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ π‘‹πΆπ‘Œ) β†’ 𝑋 ∈ 𝐡)
3 simplr2 1216 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ π‘‹πΆπ‘Œ) β†’ π‘Œ ∈ 𝐡)
4 simpr 485 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ π‘‹πΆπ‘Œ) β†’ π‘‹πΆπ‘Œ)
5 cvrletr.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
6 cvrletr.s . . . . 5 < = (ltβ€˜πΎ)
7 cvrletr.c . . . . 5 𝐢 = ( β‹– β€˜πΎ)
85, 6, 7cvrlt 38135 . . . 4 (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ π‘‹πΆπ‘Œ) β†’ 𝑋 < π‘Œ)
91, 2, 3, 4, 8syl31anc 1373 . . 3 (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ π‘‹πΆπ‘Œ) β†’ 𝑋 < π‘Œ)
10 cvrletr.l . . . . 5 ≀ = (leβ€˜πΎ)
115, 10, 6pltletr 18295 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝑋 < π‘Œ ∧ π‘Œ ≀ 𝑍) β†’ 𝑋 < 𝑍))
1211adantr 481 . . 3 (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ π‘‹πΆπ‘Œ) β†’ ((𝑋 < π‘Œ ∧ π‘Œ ≀ 𝑍) β†’ 𝑋 < 𝑍))
139, 12mpand 693 . 2 (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ π‘‹πΆπ‘Œ) β†’ (π‘Œ ≀ 𝑍 β†’ 𝑋 < 𝑍))
1413expimpd 454 1 ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((π‘‹πΆπ‘Œ ∧ π‘Œ ≀ 𝑍) β†’ 𝑋 < 𝑍))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   class class class wbr 5148  β€˜cfv 6543  Basecbs 17143  lecple 17203  Posetcpo 18259  ltcplt 18260   β‹– ccvr 38127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-proset 18247  df-poset 18265  df-plt 18282  df-covers 38131
This theorem is referenced by: (None)
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