Theorem shle0 29225

Description: No subspace is smaller than the zero subspace. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
shle0	⊢ (𝐴 ∈ Sℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ))

Proof of Theorem shle0

Step	Hyp	Ref	Expression
1		sh0le 29223	. . 3 ⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴)
2	1	biantrud 535	. 2 ⊢ (𝐴 ∈ Sℋ → (𝐴 ⊆ 0ℋ ↔ (𝐴 ⊆ 0ℋ ∧ 0ℋ ⊆ 𝐴)))
3		eqss 3930	. 2 ⊢ (𝐴 = 0ℋ ↔ (𝐴 ⊆ 0ℋ ∧ 0ℋ ⊆ 𝐴))
4	2, 3	syl6bbr 292	1 ⊢ (𝐴 ∈ Sℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ⊆ wss 3881   S_ℋ csh 28711  0_ℋc0h 28718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-hilex 28782

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-sh 28990  df-ch0 29036

This theorem is referenced by:  chle0  29226  shne0i  29231  shs00i  29233  cdj3lem1  30217