shsspwh - Hilbert Space Explorer

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Theorem shsspwh 28658

Description: Subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)

**Assertion**
Ref	Expression
shsspwh	⊢ S_ℋ ⊆ 𝒫 ℋ

**Proof of Theorem shsspwh**
Step	Hyp	Ref	Expression
1		pwuni 4696	. 2 ⊢ S_ℋ ⊆ 𝒫 ∪ S_ℋ
2		helsh 28657	. . . 4 ⊢ ℋ ∈ S_ℋ
3		shss 28622	. . . . 5 ⊢ (𝑥 ∈ S_ℋ → 𝑥 ⊆ ℋ)
4	3	rgen 3131	. . . 4 ⊢ ∀𝑥 ∈ S_ℋ 𝑥 ⊆ ℋ
5		ssunieq 4694	. . . 4 ⊢ (( ℋ ∈ S_ℋ ∧ ∀𝑥 ∈ S_ℋ 𝑥 ⊆ ℋ) → ℋ = ∪ S_ℋ )
6	2, 4, 5	mp2an 685	. . 3 ⊢ ℋ = ∪ S_ℋ
7	6	pweqi 4382	. 2 ⊢ 𝒫 ℋ = 𝒫 ∪ S_ℋ
8	1, 7	sseqtr4i 3863	1 ⊢ S_ℋ ⊆ 𝒫 ℋ

Colors of variables: wff setvar class

Syntax hints: = wceq 1658 ∈ wcel 2166 ∀wral 3117 ⊆ wss 3798 𝒫 cpw 4378 ∪ cuni 4658 ℋchba 28331 S_ℋ csh 28340

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-1cn 10310 ax-addcl 10312 ax-hilex 28411 ax-hfvadd 28412 ax-hv0cl 28415 ax-hfvmul 28417

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-map 8124 df-nn 11351 df-hlim 28384 df-sh 28619 df-ch 28633

This theorem is referenced by: chsspwh 28659 shsupunss 28760