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Theorem hl2at 38878
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 2728 . . 3 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 eqid 2728 . . 3 (ltβ€˜πΎ) = (ltβ€˜πΎ)
3 eqid 2728 . . 3 (0.β€˜πΎ) = (0.β€˜πΎ)
4 eqid 2728 . . 3 (1.β€˜πΎ) = (1.β€˜πΎ)
51, 2, 3, 4hlhgt2 38862 . 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 38834 . . . . . . . 8 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
87adantr 480 . . . . . . 7 ((𝐾 ∈ HL ∧ π‘₯ ∈ (Baseβ€˜πΎ)) β†’ 𝐾 ∈ OP)
91, 3op0cl 38656 . . . . . . 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 2728 . . . . . . 7 (leβ€˜πΎ) = (leβ€˜πΎ)
13 hl2atom.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
141, 12, 2, 13hlrelat1 38873 . . . . . 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 1369 . . . . 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 38657 . . . . . . 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 38873 . . . . . 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 1459 . . . . 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 3223 . . . . 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 5168 . . . . . . . 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 3081 . . . . . 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 3081 . . . . 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 3152 . 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 1534   ∈ wcel 2099   β‰  wne 2937  βˆƒwrex 3067   class class class wbr 5148  β€˜cfv 6548  Basecbs 17180  lecple 17240  ltcplt 18300  0.cp0 18415  1.cp1 18416  OPcops 38644  Atomscatm 38735  HLchlt 38822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-proset 18287  df-poset 18305  df-plt 18322  df-lub 18338  df-glb 18339  df-join 18340  df-meet 18341  df-p0 18417  df-p1 18418  df-lat 18424  df-clat 18491  df-oposet 38648  df-ol 38650  df-oml 38651  df-covers 38738  df-ats 38739  df-atl 38770  df-cvlat 38794  df-hlat 38823
This theorem is referenced by:  atex  38879
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