shsspwh - Hilbert Space Explorer

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Theorem shsspwh 29023

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 4875	. 2 ⊢ S_ℋ ⊆ 𝒫 ∪ S_ℋ
2		helsh 29022	. . . 4 ⊢ ℋ ∈ S_ℋ
3		shss 28987	. . . . 5 ⊢ (𝑥 ∈ S_ℋ → 𝑥 ⊆ ℋ)
4	3	rgen 3148	. . . 4 ⊢ ∀𝑥 ∈ S_ℋ 𝑥 ⊆ ℋ
5		ssunieq 4873	. . . 4 ⊢ (( ℋ ∈ S_ℋ ∧ ∀𝑥 ∈ S_ℋ 𝑥 ⊆ ℋ) → ℋ = ∪ S_ℋ )
6	2, 4, 5	mp2an 690	. . 3 ⊢ ℋ = ∪ S_ℋ
7	6	pweqi 4557	. 2 ⊢ 𝒫 ℋ = 𝒫 ∪ S_ℋ
8	1, 7	sseqtrri 4004	1 ⊢ S_ℋ ⊆ 𝒫 ℋ

Colors of variables: wff setvar class

Syntax hints: = wceq 1537 ∈ wcel 2114 ∀wral 3138 ⊆ wss 3936 𝒫 cpw 4539 ∪ cuni 4838 ℋchba 28696 S_ℋ csh 28705

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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-1cn 10595 ax-addcl 10597 ax-hilex 28776 ax-hfvadd 28777 ax-hv0cl 28780 ax-hfvmul 28782

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-map 8408 df-nn 11639 df-hlim 28749 df-sh 28984 df-ch 28998

This theorem is referenced by: chsspwh 29024 shsupunss 29123