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Theorem cvrletrN 36391
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 1209 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑌) → 𝑋𝐵)
3 simplr2 1210 . . . 4 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑌) → 𝑌𝐵)
4 simpr 487 . . . 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 36388 . . . 4 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
91, 2, 3, 4, 8syl31anc 1367 . . 3 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
10 cvrletr.l . . . . 5 = (le‘𝐾)
115, 10, 6pltletr 17573 . . . 4 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 < 𝑌𝑌 𝑍) → 𝑋 < 𝑍))
1211adantr 483 . . 3 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑌𝑌 𝑍) → 𝑋 < 𝑍))
139, 12mpand 693 . 2 (((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑌) → (𝑌 𝑍𝑋 < 𝑍))
1413expimpd 456 1 ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋𝐶𝑌𝑌 𝑍) → 𝑋 < 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1081   = wceq 1530  wcel 2107   class class class wbr 5057  cfv 6348  Basecbs 16475  lecple 16564  Posetcpo 17542  ltcplt 17543  ccvr 36380
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-proset 17530  df-poset 17548  df-plt 17560  df-covers 36384
This theorem is referenced by: (None)
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