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Theorem chnlen0 31202
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 31199 . . 3 (𝐵C → 0𝐵)
2 sseq1 4002 . . 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 3943   C cch 30687  0c0h 30693
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 2697  ax-sep 5292  ax-hilex 30757
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-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fv 6544  df-ov 7407  df-sh 30965  df-ch 30979  df-ch0 31011
This theorem is referenced by: (None)
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