shsspwh - Hilbert Space Explorer

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Theorem shsspwh 31332

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 4889	. 2 ⊢ S_ℋ ⊆ 𝒫 ∪ S_ℋ
2		helsh 31331	. . . 4 ⊢ ℋ ∈ S_ℋ
3		shss 31296	. . . . 5 ⊢ (𝑥 ∈ S_ℋ → 𝑥 ⊆ ℋ)
4	3	rgen 3054	. . . 4 ⊢ ∀𝑥 ∈ S_ℋ 𝑥 ⊆ ℋ
5		ssunieq 4887	. . . 4 ⊢ (( ℋ ∈ S_ℋ ∧ ∀𝑥 ∈ S_ℋ 𝑥 ⊆ ℋ) → ℋ = ∪ S_ℋ )
6	2, 4, 5	mp2an 693	. . 3 ⊢ ℋ = ∪ S_ℋ
7	6	pweqi 4558	. 2 ⊢ 𝒫 ℋ = 𝒫 ∪ S_ℋ
8	1, 7	sseqtrri 3972	1 ⊢ S_ℋ ⊆ 𝒫 ℋ

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ∈ wcel 2114 ∀wral 3052 ⊆ wss 3890 𝒫 cpw 4542 ∪ cuni 4851 ℋchba 31005 S_ℋ csh 31014

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-1cn 11087 ax-addcl 11089 ax-hilex 31085 ax-hfvadd 31086 ax-hv0cl 31089 ax-hfvmul 31091

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-map 8768 df-nn 12166 df-hlim 31058 df-sh 31293 df-ch 31307

This theorem is referenced by: chsspwh 31333 shsupunss 31432