chle0i - Hilbert Space Explorer

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Theorem chle0i 31334

Description: No Hilbert closed subspace is smaller than zero. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)

**Hypothesis**
Ref	Expression
ch0le.1	⊢ 𝐴 ∈ C_ℋ

**Assertion**
Ref	Expression
chle0i	⊢ (𝐴 ⊆ 0_ℋ ↔ 𝐴 = 0_ℋ)

**Proof of Theorem chle0i**
Step	Hyp	Ref	Expression
1		ch0le.1	. 2 ⊢ 𝐴 ∈ C_ℋ
2		chle0 31325	. 2 ⊢ (𝐴 ∈ C_ℋ → (𝐴 ⊆ 0_ℋ ↔ 𝐴 = 0_ℋ))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ⊆ 0_ℋ ↔ 𝐴 = 0_ℋ)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 C_ℋ cch 30811 0_ℋc0h 30817

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-hilex 30881

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5684 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fv 6557 df-ov 7422 df-sh 31089 df-ch 31103 df-ch0 31135

This theorem is referenced by: chj00i 31369 chsup0 31430 spansnm0i 31532 largei 32149

	
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