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Theorem cvrletrN 34379
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 789 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑌) → 𝐾 ∈ Poset)
2 simplr1 1101 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑌) → 𝑋𝐵)
3 simplr2 1102 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑌) → 𝑌𝐵)
4 simpr 477 . . . 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 34376 . . . 4 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
91, 2, 3, 4, 8syl31anc 1327 . . 3 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
10 cvrletr.l . . . . 5 = (le‘𝐾)
115, 10, 6pltletr 16952 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 𝑍) → 𝑋 < 𝑍))
1211adantr 481 . . 3 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑌𝑌 𝑍) → 𝑋 < 𝑍))
139, 12mpand 710 . 2 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑌) → (𝑌 𝑍𝑋 < 𝑍))
1413expimpd 628 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋𝐶𝑌𝑌 𝑍) → 𝑋 < 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988   class class class wbr 4644  cfv 5876  Basecbs 15838  lecple 15929  Posetcpo 16921  ltcplt 16922  ccvr 34368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-preset 16909  df-poset 16927  df-plt 16939  df-covers 34372
This theorem is referenced by: (None)
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