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Theorem hl2at 38780
Description: A Hilbert lattice has at least 2 atoms. (Contributed by NM, 5-Dec-2011.)
Hypothesis
Ref Expression
hl2atom.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
hl2at (𝐾 ∈ HL β†’ βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 𝑝 β‰  π‘ž)
Distinct variable groups:   π‘ž,𝑝,𝐴   𝐾,𝑝,π‘ž

Proof of Theorem hl2at
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 eqid 2724 . . 3 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 eqid 2724 . . 3 (ltβ€˜πΎ) = (ltβ€˜πΎ)
3 eqid 2724 . . 3 (0.β€˜πΎ) = (0.β€˜πΎ)
4 eqid 2724 . . 3 (1.β€˜πΎ) = (1.β€˜πΎ)
51, 2, 3, 4hlhgt2 38764 . 2 (𝐾 ∈ HL β†’ βˆƒπ‘₯ ∈ (Baseβ€˜πΎ)((0.β€˜πΎ)(ltβ€˜πΎ)π‘₯ ∧ π‘₯(ltβ€˜πΎ)(1.β€˜πΎ)))
6 simpl 482 . . . . . 6 ((𝐾 ∈ HL ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ 𝐾 ∈ HL)
7 hlop 38736 . . . . . . . 8 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
87adantr 480 . . . . . . 7 ((𝐾 ∈ HL ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ 𝐾 ∈ OP)
91, 3op0cl 38558 . . . . . . 7 (𝐾 ∈ OP β†’ (0.β€˜πΎ) ∈ (Baseβ€˜πΎ))
108, 9syl 17 . . . . . 6 ((𝐾 ∈ HL ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ (0.β€˜πΎ) ∈ (Baseβ€˜πΎ))
11 simpr 484 . . . . . 6 ((𝐾 ∈ HL ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ π‘₯ ∈ (Baseβ€˜πΎ))
12 eqid 2724 . . . . . . 7 (leβ€˜πΎ) = (leβ€˜πΎ)
13 hl2atom.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
141, 12, 2, 13hlrelat1 38775 . . . . . 6 ((𝐾 ∈ HL ∧ (0.β€˜πΎ) ∈ (Baseβ€˜πΎ) ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ ((0.β€˜πΎ)(ltβ€˜πΎ)π‘₯ β†’ βˆƒπ‘ ∈ 𝐴 (Β¬ 𝑝(leβ€˜πΎ)(0.β€˜πΎ) ∧ 𝑝(leβ€˜πΎ)π‘₯)))
156, 10, 11, 14syl3anc 1368 . . . . 5 ((𝐾 ∈ HL ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ ((0.β€˜πΎ)(ltβ€˜πΎ)π‘₯ β†’ βˆƒπ‘ ∈ 𝐴 (Β¬ 𝑝(leβ€˜πΎ)(0.β€˜πΎ) ∧ 𝑝(leβ€˜πΎ)π‘₯)))
161, 4op1cl 38559 . . . . . . 7 (𝐾 ∈ OP β†’ (1.β€˜πΎ) ∈ (Baseβ€˜πΎ))
178, 16syl 17 . . . . . 6 ((𝐾 ∈ HL ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ (1.β€˜πΎ) ∈ (Baseβ€˜πΎ))
181, 12, 2, 13hlrelat1 38775 . . . . . 6 ((𝐾 ∈ HL ∧ π‘₯ ∈ (Baseβ€˜πΎ) ∧ (1.β€˜πΎ) ∈ (Baseβ€˜πΎ)) β†’ (π‘₯(ltβ€˜πΎ)(1.β€˜πΎ) β†’ βˆƒπ‘ž ∈ 𝐴 (Β¬ π‘ž(leβ€˜πΎ)π‘₯ ∧ π‘ž(leβ€˜πΎ)(1.β€˜πΎ))))
1917, 18mpd3an3 1458 . . . . 5 ((𝐾 ∈ HL ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ (π‘₯(ltβ€˜πΎ)(1.β€˜πΎ) β†’ βˆƒπ‘ž ∈ 𝐴 (Β¬ π‘ž(leβ€˜πΎ)π‘₯ ∧ π‘ž(leβ€˜πΎ)(1.β€˜πΎ))))
2015, 19anim12d 608 . . . 4 ((𝐾 ∈ HL ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ (((0.β€˜πΎ)(ltβ€˜πΎ)π‘₯ ∧ π‘₯(ltβ€˜πΎ)(1.β€˜πΎ)) β†’ (βˆƒπ‘ ∈ 𝐴 (Β¬ 𝑝(leβ€˜πΎ)(0.β€˜πΎ) ∧ 𝑝(leβ€˜πΎ)π‘₯) ∧ βˆƒπ‘ž ∈ 𝐴 (Β¬ π‘ž(leβ€˜πΎ)π‘₯ ∧ π‘ž(leβ€˜πΎ)(1.β€˜πΎ)))))
21 reeanv 3218 . . . . 5 (βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 ((Β¬ 𝑝(leβ€˜πΎ)(0.β€˜πΎ) ∧ 𝑝(leβ€˜πΎ)π‘₯) ∧ (Β¬ π‘ž(leβ€˜πΎ)π‘₯ ∧ π‘ž(leβ€˜πΎ)(1.β€˜πΎ))) ↔ (βˆƒπ‘ ∈ 𝐴 (Β¬ 𝑝(leβ€˜πΎ)(0.β€˜πΎ) ∧ 𝑝(leβ€˜πΎ)π‘₯) ∧ βˆƒπ‘ž ∈ 𝐴 (Β¬ π‘ž(leβ€˜πΎ)π‘₯ ∧ π‘ž(leβ€˜πΎ)(1.β€˜πΎ))))
22 nbrne2 5159 . . . . . . . 8 ((𝑝(leβ€˜πΎ)π‘₯ ∧ Β¬ π‘ž(leβ€˜πΎ)π‘₯) β†’ 𝑝 β‰  π‘ž)
2322ad2ant2lr 745 . . . . . . 7 (((Β¬ 𝑝(leβ€˜πΎ)(0.β€˜πΎ) ∧ 𝑝(leβ€˜πΎ)π‘₯) ∧ (Β¬ π‘ž(leβ€˜πΎ)π‘₯ ∧ π‘ž(leβ€˜πΎ)(1.β€˜πΎ))) β†’ 𝑝 β‰  π‘ž)
2423reximi 3076 . . . . . 6 (βˆƒπ‘ž ∈ 𝐴 ((Β¬ 𝑝(leβ€˜πΎ)(0.β€˜πΎ) ∧ 𝑝(leβ€˜πΎ)π‘₯) ∧ (Β¬ π‘ž(leβ€˜πΎ)π‘₯ ∧ π‘ž(leβ€˜πΎ)(1.β€˜πΎ))) β†’ βˆƒπ‘ž ∈ 𝐴 𝑝 β‰  π‘ž)
2524reximi 3076 . . . . 5 (βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 ((Β¬ 𝑝(leβ€˜πΎ)(0.β€˜πΎ) ∧ 𝑝(leβ€˜πΎ)π‘₯) ∧ (Β¬ π‘ž(leβ€˜πΎ)π‘₯ ∧ π‘ž(leβ€˜πΎ)(1.β€˜πΎ))) β†’ βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 𝑝 β‰  π‘ž)
2621, 25sylbir 234 . . . 4 ((βˆƒπ‘ ∈ 𝐴 (Β¬ 𝑝(leβ€˜πΎ)(0.β€˜πΎ) ∧ 𝑝(leβ€˜πΎ)π‘₯) ∧ βˆƒπ‘ž ∈ 𝐴 (Β¬ π‘ž(leβ€˜πΎ)π‘₯ ∧ π‘ž(leβ€˜πΎ)(1.β€˜πΎ))) β†’ βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 𝑝 β‰  π‘ž)
2720, 26syl6 35 . . 3 ((𝐾 ∈ HL ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ (((0.β€˜πΎ)(ltβ€˜πΎ)π‘₯ ∧ π‘₯(ltβ€˜πΎ)(1.β€˜πΎ)) β†’ βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 𝑝 β‰  π‘ž))
2827rexlimdva 3147 . 2 (𝐾 ∈ HL β†’ (βˆƒπ‘₯ ∈ (Baseβ€˜πΎ)((0.β€˜πΎ)(ltβ€˜πΎ)π‘₯ ∧ π‘₯(ltβ€˜πΎ)(1.β€˜πΎ)) β†’ βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 𝑝 β‰  π‘ž))
295, 28mpd 15 1 (𝐾 ∈ HL β†’ βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 𝑝 β‰  π‘ž)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   β‰  wne 2932  βˆƒwrex 3062   class class class wbr 5139  β€˜cfv 6534  Basecbs 17149  lecple 17209  ltcplt 18269  0.cp0 18384  1.cp1 18385  OPcops 38546  Atomscatm 38637  HLchlt 38724
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-proset 18256  df-poset 18274  df-plt 18291  df-lub 18307  df-glb 18308  df-join 18309  df-meet 18310  df-p0 18386  df-p1 18387  df-lat 18393  df-clat 18460  df-oposet 38550  df-ol 38552  df-oml 38553  df-covers 38640  df-ats 38641  df-atl 38672  df-cvlat 38696  df-hlat 38725
This theorem is referenced by:  atex  38781
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