Theorem shne0i 29227

Description: A nonzero subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
shne0.1	⊢ 𝐴 ∈ Sℋ

Assertion

Ref	Expression
shne0i	⊢ (𝐴 ≠ 0ℋ ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ)

Distinct variable group:   𝑥,𝐴

Proof of Theorem shne0i

Step	Hyp	Ref	Expression
1		df-ne 3019	. 2 ⊢ (𝐴 ≠ 0ℋ ↔ ¬ 𝐴 = 0ℋ)
2		df-rex 3146	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 0ℋ ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 0ℋ))
3		nss 4031	. . 3 ⊢ (¬ 𝐴 ⊆ 0ℋ ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 0ℋ))
4		shne0.1	. . . . 5 ⊢ 𝐴 ∈ Sℋ
5		shle0 29221	. . . . 5 ⊢ (𝐴 ∈ Sℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ))
6	4, 5	ax-mp 5	. . . 4 ⊢ (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ)
7	6	notbii 322	. . 3 ⊢ (¬ 𝐴 ⊆ 0ℋ ↔ ¬ 𝐴 = 0ℋ)
8	2, 3, 7	3bitr2ri 302	. 2 ⊢ (¬ 𝐴 = 0ℋ ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 0ℋ)
9		elch0 29033	. . . 4 ⊢ (𝑥 ∈ 0ℋ ↔ 𝑥 = 0ℎ)
10	9	necon3bbii 3065	. . 3 ⊢ (¬ 𝑥 ∈ 0ℋ ↔ 𝑥 ≠ 0ℎ)
11	10	rexbii 3249	. 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 0ℋ ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ)
12	1, 8, 11	3bitri 299	1 ⊢ (𝐴 ≠ 0ℋ ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ)

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ≠ wne 3018  ∃wrex 3141   ⊆ wss 3938  0_ℎc0v 28703   S_ℋ csh 28707  0_ℋc0h 28714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-hilex 28778  ax-hv0cl 28782

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-xp 5563  df-cnv 5565  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-sh 28986  df-ch0 29032

This theorem is referenced by:  chne0i  29232  shatomici  30137