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Theorem cvrletrN 37948
Description: Property of an element above a covering. (Contributed by NM, 7-Dec-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cvrletr.b 𝐵 = (Base‘𝐾)
cvrletr.l = (le‘𝐾)
cvrletr.s < = (lt‘𝐾)
cvrletr.c 𝐶 = ( ⋖ ‘𝐾)
Assertion
Ref Expression
cvrletrN ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋𝐶𝑌𝑌 𝑍) → 𝑋 < 𝑍))

Proof of Theorem cvrletrN
StepHypRef Expression
1 simpll 765 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑌) → 𝐾 ∈ Poset)
2 simplr1 1215 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑌) → 𝑋𝐵)
3 simplr2 1216 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑌) → 𝑌𝐵)
4 simpr 485 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑌) → 𝑋𝐶𝑌)
5 cvrletr.b . . . . 5 𝐵 = (Base‘𝐾)
6 cvrletr.s . . . . 5 < = (lt‘𝐾)
7 cvrletr.c . . . . 5 𝐶 = ( ⋖ ‘𝐾)
85, 6, 7cvrlt 37945 . . . 4 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
91, 2, 3, 4, 8syl31anc 1373 . . 3 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
10 cvrletr.l . . . . 5 = (le‘𝐾)
115, 10, 6pltletr 18278 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 𝑍) → 𝑋 < 𝑍))
1211adantr 481 . . 3 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑌𝑌 𝑍) → 𝑋 < 𝑍))
139, 12mpand 693 . 2 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑌) → (𝑌 𝑍𝑋 < 𝑍))
1413expimpd 454 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋𝐶𝑌𝑌 𝑍) → 𝑋 < 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106   class class class wbr 5141  cfv 6532  Basecbs 17126  lecple 17186  Posetcpo 18242  ltcplt 18243  ccvr 37937
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 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  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-iota 6484  df-fun 6534  df-fv 6540  df-proset 18230  df-poset 18248  df-plt 18265  df-covers 37941
This theorem is referenced by: (None)
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