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Theorem intnatN 36423
Description: If the intersection with a non-majorizing element is an atom, the intersecting element is not an atom. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
intnat.b 𝐵 = (Base‘𝐾)
intnat.l = (le‘𝐾)
intnat.m = (meet‘𝐾)
intnat.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
intnatN (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (¬ 𝑌 𝑋 ∧ (𝑋 𝑌) ∈ 𝐴)) → ¬ 𝑌𝐴)

Proof of Theorem intnatN
StepHypRef Expression
1 hlatl 36376 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
213ad2ant1 1125 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ AtLat)
32ad2antrr 722 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝐾 ∈ AtLat)
4 eqid 2818 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
5 intnat.a . . . . . 6 𝐴 = (Atoms‘𝐾)
64, 5atn0 36324 . . . . 5 ((𝐾 ∈ AtLat ∧ (𝑋 𝑌) ∈ 𝐴) → (𝑋 𝑌) ≠ (0.‘𝐾))
73, 6sylancom 588 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → (𝑋 𝑌) ≠ (0.‘𝐾))
87ex 413 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) → ((𝑋 𝑌) ∈ 𝐴 → (𝑋 𝑌) ≠ (0.‘𝐾)))
9 simpll1 1204 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝐾 ∈ HL)
109hllatd 36380 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝐾 ∈ Lat)
11 simpll2 1205 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝑋𝐵)
12 simpll3 1206 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝑌𝐵)
13 intnat.b . . . . . . . 8 𝐵 = (Base‘𝐾)
14 intnat.m . . . . . . . 8 = (meet‘𝐾)
1513, 14latmcom 17673 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))
1610, 11, 12, 15syl3anc 1363 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → (𝑋 𝑌) = (𝑌 𝑋))
17 simplr 765 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → ¬ 𝑌 𝑋)
189, 1syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝐾 ∈ AtLat)
19 simpr 485 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝑌𝐴)
20 intnat.l . . . . . . . . 9 = (le‘𝐾)
2113, 20, 14, 4, 5atnle 36333 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑌𝐴𝑋𝐵) → (¬ 𝑌 𝑋 ↔ (𝑌 𝑋) = (0.‘𝐾)))
2218, 19, 11, 21syl3anc 1363 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → (¬ 𝑌 𝑋 ↔ (𝑌 𝑋) = (0.‘𝐾)))
2317, 22mpbid 233 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → (𝑌 𝑋) = (0.‘𝐾))
2416, 23eqtrd 2853 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → (𝑋 𝑌) = (0.‘𝐾))
2524ex 413 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) → (𝑌𝐴 → (𝑋 𝑌) = (0.‘𝐾)))
2625necon3ad 3026 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) → ((𝑋 𝑌) ≠ (0.‘𝐾) → ¬ 𝑌𝐴))
278, 26syld 47 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) → ((𝑋 𝑌) ∈ 𝐴 → ¬ 𝑌𝐴))
2827impr 455 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (¬ 𝑌 𝑋 ∧ (𝑋 𝑌) ∈ 𝐴)) → ¬ 𝑌𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013   class class class wbr 5057  cfv 6348  (class class class)co 7145  Basecbs 16471  lecple 16560  meetcmee 17543  0.cp0 17635  Latclat 17643  Atomscatm 36279  AtLatcal 36280  HLchlt 36366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  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-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-proset 17526  df-poset 17544  df-plt 17556  df-lub 17572  df-glb 17573  df-join 17574  df-meet 17575  df-p0 17637  df-lat 17644  df-covers 36282  df-ats 36283  df-atl 36314  df-cvlat 36338  df-hlat 36367
This theorem is referenced by: (None)
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