Theorem shne0 9371

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 1587	. 2 |- (A =/= 0H <-> -. A = 0H)
2		df-rex 1650	. . 3 |- (E.x e. A -. x e. 0H <-> E.x(x e. A /\ -. x e. 0H))
3		nss 2113	. . 3 |- (-. A (_ 0H <-> E.x(x e. A /\ -. x e. 0H))
4		shne0.1	. . . . 5 |- A e. SH
5		shle0t 9366	. . . . 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 9126	. . . 4 |- (x e. 0H <-> x = 0h)
10	9	necon3bbii 1597	. . 3 |- (-. x e. 0H <-> x =/= 0h)
11	10	rexbii 1668	. 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 956   e. wcel 958  E.wex 980   =/= wne 1585  E.wrex 1646   (_ wss 2047  0hc0v 8791  SHcsh 8797  0Hc0h 8804

This theorem is referenced by:  chne0 9376  shatomic 10285

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-hilex 8869  ax-hv0cl 8873

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-un 2050  df-in 2051  df-ss 2053  df-sn 2412  df-pr 2413  df-sh 9076  df-ch0 9125

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