Theorem chle0 28302

Description: No Hilbert lattice element is smaller than zero. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)

Assertion

Ref	Expression
chle0	|- ( A e. CH -> ( A C_ 0H <-> A = 0H ) )

Proof of Theorem chle0

Step	Hyp	Ref	Expression
1		chsh 28081	. 2 |- ( A e. CH -> A e. SH )
2		shle0 28301	. 2 |- ( A e. SH -> ( A C_ 0H <-> A = 0H ) )
3	1, 2	syl 17	1 |- ( A e. CH -> ( A C_ 0H <-> A = 0H ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   C_ wss 3574   SHcsh 27785   CHcch 27786   0Hc0h 27792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-hilex 27856

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fv 5896  df-ov 6653  df-sh 28064  df-ch 28078  df-ch0 28110

This theorem is referenced by:  chle0i  28311  chssoc  28355  hatomistici  29221  atcvat4i  29256