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Theorem hl2at 37914
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 2733 . . 3 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 eqid 2733 . . 3 (ltβ€˜πΎ) = (ltβ€˜πΎ)
3 eqid 2733 . . 3 (0.β€˜πΎ) = (0.β€˜πΎ)
4 eqid 2733 . . 3 (1.β€˜πΎ) = (1.β€˜πΎ)
51, 2, 3, 4hlhgt2 37898 . 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 484 . . . . . 6 ((𝐾 ∈ HL ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ 𝐾 ∈ HL)
7 hlop 37870 . . . . . . . 8 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
87adantr 482 . . . . . . 7 ((𝐾 ∈ HL ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ 𝐾 ∈ OP)
91, 3op0cl 37692 . . . . . . 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 486 . . . . . 6 ((𝐾 ∈ HL ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ π‘₯ ∈ (Baseβ€˜πΎ))
12 eqid 2733 . . . . . . 7 (leβ€˜πΎ) = (leβ€˜πΎ)
13 hl2atom.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
141, 12, 2, 13hlrelat1 37909 . . . . . 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 1372 . . . . 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 37693 . . . . . . 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 37909 . . . . . 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 1463 . . . . 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 610 . . . 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 3216 . . . . 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 5126 . . . . . . . 8 ((𝑝(leβ€˜πΎ)π‘₯ ∧ Β¬ π‘ž(leβ€˜πΎ)π‘₯) β†’ 𝑝 β‰  π‘ž)
2322ad2ant2lr 747 . . . . . . 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 3084 . . . . . 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 3084 . . . . 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 3149 . 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 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆƒwrex 3070   class class class wbr 5106  β€˜cfv 6497  Basecbs 17088  lecple 17145  ltcplt 18202  0.cp0 18317  1.cp1 18318  OPcops 37680  Atomscatm 37771  HLchlt 37858
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-ov 7361  df-oprab 7362  df-proset 18189  df-poset 18207  df-plt 18224  df-lub 18240  df-glb 18241  df-join 18242  df-meet 18243  df-p0 18319  df-p1 18320  df-lat 18326  df-clat 18393  df-oposet 37684  df-ol 37686  df-oml 37687  df-covers 37774  df-ats 37775  df-atl 37806  df-cvlat 37830  df-hlat 37859
This theorem is referenced by:  atex  37915
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