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Theorem intnatN 35214
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 35168 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
213ad2ant1 1128 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ AtLat)
32ad2antrr 764 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → 𝐾 ∈ AtLat)
4 eqid 2760 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
5 intnat.a . . . . . 6 𝐴 = (Atoms‘𝐾)
64, 5atn0 35116 . . . . 5 ((𝐾 ∈ AtLat ∧ (𝑋 𝑌) ∈ 𝐴) → (𝑋 𝑌) ≠ (0.‘𝐾))
73, 6sylancom 704 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → (𝑋 𝑌) ≠ (0.‘𝐾))
87ex 449 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) → ((𝑋 𝑌) ∈ 𝐴 → (𝑋 𝑌) ≠ (0.‘𝐾)))
9 simpll1 1255 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝐾 ∈ HL)
10 hllat 35171 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ Lat)
119, 10syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝐾 ∈ Lat)
12 simpll2 1257 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝑋𝐵)
13 simpll3 1259 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝑌𝐵)
14 intnat.b . . . . . . . 8 𝐵 = (Base‘𝐾)
15 intnat.m . . . . . . . 8 = (meet‘𝐾)
1614, 15latmcom 17296 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))
1711, 12, 13, 16syl3anc 1477 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → (𝑋 𝑌) = (𝑌 𝑋))
18 simplr 809 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → ¬ 𝑌 𝑋)
199, 1syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝐾 ∈ AtLat)
20 simpr 479 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → 𝑌𝐴)
21 intnat.l . . . . . . . . 9 = (le‘𝐾)
2214, 21, 15, 4, 5atnle 35125 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑌𝐴𝑋𝐵) → (¬ 𝑌 𝑋 ↔ (𝑌 𝑋) = (0.‘𝐾)))
2319, 20, 12, 22syl3anc 1477 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → (¬ 𝑌 𝑋 ↔ (𝑌 𝑋) = (0.‘𝐾)))
2418, 23mpbid 222 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → (𝑌 𝑋) = (0.‘𝐾))
2517, 24eqtrd 2794 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) ∧ 𝑌𝐴) → (𝑋 𝑌) = (0.‘𝐾))
2625ex 449 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) → (𝑌𝐴 → (𝑋 𝑌) = (0.‘𝐾)))
2726necon3ad 2945 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) → ((𝑋 𝑌) ≠ (0.‘𝐾) → ¬ 𝑌𝐴))
288, 27syld 47 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ¬ 𝑌 𝑋) → ((𝑋 𝑌) ∈ 𝐴 → ¬ 𝑌𝐴))
2928impr 650 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (¬ 𝑌 𝑋 ∧ (𝑋 𝑌) ∈ 𝐴)) → ¬ 𝑌𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932   class class class wbr 4804  cfv 6049  (class class class)co 6814  Basecbs 16079  lecple 16170  meetcmee 17166  0.cp0 17258  Latclat 17266  Atomscatm 35071  AtLatcal 35072  HLchlt 35158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-preset 17149  df-poset 17167  df-plt 17179  df-lub 17195  df-glb 17196  df-join 17197  df-meet 17198  df-p0 17260  df-lat 17267  df-covers 35074  df-ats 35075  df-atl 35106  df-cvlat 35130  df-hlat 35159
This theorem is referenced by: (None)
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