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Theorem atncvrN 37823
Description: Two atoms cannot satisfy the covering relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
atncvr.c 𝐢 = ( β‹– β€˜πΎ)
atncvr.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
atncvrN ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ Β¬ 𝑃𝐢𝑄)

Proof of Theorem atncvrN
StepHypRef Expression
1 eqid 2733 . . . 4 (0.β€˜πΎ) = (0.β€˜πΎ)
2 atncvr.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
31, 2atn0 37816 . . 3 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ 𝑃 β‰  (0.β€˜πΎ))
433adant3 1133 . 2 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ 𝑃 β‰  (0.β€˜πΎ))
5 eqid 2733 . . . . 5 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
65, 2atbase 37797 . . . 4 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
7 eqid 2733 . . . . 5 (leβ€˜πΎ) = (leβ€˜πΎ)
8 atncvr.c . . . . 5 𝐢 = ( β‹– β€˜πΎ)
95, 7, 1, 8, 2atcvreq0 37822 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ 𝐴) β†’ (𝑃𝐢𝑄 ↔ 𝑃 = (0.β€˜πΎ)))
106, 9syl3an2 1165 . . 3 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃𝐢𝑄 ↔ 𝑃 = (0.β€˜πΎ)))
1110necon3bbid 2978 . 2 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (Β¬ 𝑃𝐢𝑄 ↔ 𝑃 β‰  (0.β€˜πΎ)))
124, 11mpbird 257 1 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ Β¬ 𝑃𝐢𝑄)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940   class class class wbr 5106  β€˜cfv 6497  Basecbs 17088  lecple 17145  0.cp0 18317   β‹– ccvr 37770  Atomscatm 37771  AtLatcal 37772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-proset 18189  df-poset 18207  df-plt 18224  df-glb 18241  df-p0 18319  df-lat 18326  df-covers 37774  df-ats 37775  df-atl 37806
This theorem is referenced by: (None)
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