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Theorem chnlen0 31272
Description: A Hilbert lattice element that is not a subset of another is nonzero. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
chnlen0 (𝐵C → (¬ 𝐴𝐵 → ¬ 𝐴 = 0))

Proof of Theorem chnlen0
StepHypRef Expression
1 ch0le 31269 . . 3 (𝐵C → 0𝐵)
2 sseq1 4005 . . 3 (𝐴 = 0 → (𝐴𝐵 ↔ 0𝐵))
31, 2syl5ibrcom 246 . 2 (𝐵C → (𝐴 = 0𝐴𝐵))
43con3d 152 1 (𝐵C → (¬ 𝐴𝐵 → ¬ 𝐴 = 0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1533  wcel 2098  wss 3947   C cch 30757  0c0h 30763
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-ext 2698  ax-sep 5301  ax-hilex 30827
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-xp 5686  df-cnv 5688  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fv 6559  df-ov 7427  df-sh 31035  df-ch 31049  df-ch0 31081
This theorem is referenced by: (None)
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