Theorem shne0 9366

Description: A non-zero subspace has a non-zero vector.

Hypothesis

Ref	Expression
shne0.1	|- A e. SH

Assertion

Ref	Expression
shne0	|- (A =/= 0H <-> E.x e. A x =/= 0h)

Distinct variable group:   x,A

Proof of Theorem shne0

Step	Hyp	Ref	Expression
1		df-ne 1590	. 2 |- (A =/= 0H <-> -. A = 0H)
2		df-rex 1653	. . 3 |- (E.x e. A -. x e. 0H <-> E.x(x e. A /\ -. x e. 0H))
3		nss 2116	. . 3 |- (-. A (_ 0H <-> E.x(x e. A /\ -. x e. 0H))
4		shne0.1	. . . . 5 |- A e. SH
5		shle0t 9361	. . . . 5 |- (A e. SH -> (A (_ 0H <-> A = 0H))
6	4, 5	ax-mp 7	. . . 4 |- (A (_ 0H <-> A = 0H)
7	6	negbii 187	. . 3 |- (-. A (_ 0H <-> -. A = 0H)
8	2, 3, 7	3bitr2r 180	. 2 |- (-. A = 0H <-> E.x e. A -. x e. 0H)
9		elch0 9121	. . . 4 |- (x e. 0H <-> x = 0h)
10	9	necon3bbii 1600	. . 3 |- (-. x e. 0H <-> x =/= 0h)
11	10	rexbii 1671	. 2 |- (E.x e. A -. x e. 0H <-> E.x e. A x =/= 0h)
12	1, 8, 11	3bitr 177	1 |- (A =/= 0H <-> E.x e. A x =/= 0h)

Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982   =/= wne 1588  E.wrex 1649   (_ wss 2050  0hc0v 8786  SHcsh 8792  0Hc0h 8799

This theorem is referenced by:  chne0 9371  shatomic 10280

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-hilex 8864  ax-hv0cl 8868

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-un 2053  df-in 2054  df-ss 2056  df-sn 2416  df-pr 2417  df-sh 9071  df-ch0 9120

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