Theorem shne0 5372

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

Hypothesis

Ref	Expression
sh0le.1	⊢ A ∈ Sℋ

Assertion

Ref	Expression
shne0	⊢ (¬ A = 0ℋ ↔ ∃x ∈ A ¬ x = 0v)

Distinct variable group(s):   x,A

Proof of Theorem shne0

Step	Hyp	Ref	Expression
1		sh0le.1	. . . . . 6 ⊢ A ∈ Sℋ
2		shle0t 5367	. . . . . 6 ⊢ (A ∈ Sℋ → (A ⊆ 0ℋ ↔ A = 0ℋ))
3	1, 2	ax-mp 6	. . . . 5 ⊢ (A ⊆ 0ℋ ↔ A = 0ℋ)
4	3	negbii 162	. . . 4 ⊢ (¬ A ⊆ 0ℋ ↔ ¬ A = 0ℋ)
5		nss 1550	. . . 4 ⊢ (¬ A ⊆ 0ℋ ↔ ∃x(x ∈ A ∧ ¬ x ∈ 0ℋ))
6	4, 5	bitr3 153	. . 3 ⊢ (¬ A = 0ℋ ↔ ∃x(x ∈ A ∧ ¬ x ∈ 0ℋ))
7		df-rex 1206	. . 3 ⊢ (∃x ∈ A ¬ x ∈ 0ℋ ↔ ∃x(x ∈ A ∧ ¬ x ∈ 0ℋ))
8	6, 7	bitr4 154	. 2 ⊢ (¬ A = 0ℋ ↔ ∃x ∈ A ¬ x ∈ 0ℋ)
9		elch0 5158	. . . 4 ⊢ (x ∈ 0ℋ ↔ x = 0v)
10	9	negbii 162	. . 3 ⊢ (¬ x ∈ 0ℋ ↔ ¬ x = 0v)
11	10	birex 1224	. 2 ⊢ (∃x ∈ A ¬ x ∈ 0ℋ ↔ ∃x ∈ A ¬ x = 0v)
12	8, 11	bitr 151	1 ⊢ (¬ A = 0ℋ ↔ ∃x ∈ A ¬ x = 0v)

Syntax hints:  ¬ wn 1   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∃wrex 1202   ⊆ wss 1487  0_vc0v 4961   S_ℋ csh 4967  0_ℋc0h 4974

This theorem is referenced by:  chne0 5375  shatomic 5753

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-hilex 4983  ax-hvzercl 4987

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-un 1490  df-in 1491  df-ss 1492  df-sn 1811  df-pr 1812  df-sh 5114  df-ch0 5157

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