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Mirrors > Home > HSE Home > Th. List > shsupunss | Structured version Visualization version GIF version |
Description: The union of a set of subspaces is smaller than its supremum. (Contributed by NM, 26-Nov-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shsupunss | ⊢ (𝐴 ⊆ Sℋ → ∪ 𝐴 ⊆ (span‘∪ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shsspwh 28404 | . . . . 5 ⊢ Sℋ ⊆ 𝒫 ℋ | |
2 | sstr 3744 | . . . . 5 ⊢ ((𝐴 ⊆ Sℋ ∧ Sℋ ⊆ 𝒫 ℋ) → 𝐴 ⊆ 𝒫 ℋ) | |
3 | 1, 2 | mpan2 709 | . . . 4 ⊢ (𝐴 ⊆ Sℋ → 𝐴 ⊆ 𝒫 ℋ) |
4 | 3 | unissd 4606 | . . 3 ⊢ (𝐴 ⊆ Sℋ → ∪ 𝐴 ⊆ ∪ 𝒫 ℋ) |
5 | unipw 5059 | . . 3 ⊢ ∪ 𝒫 ℋ = ℋ | |
6 | 4, 5 | syl6sseq 3784 | . 2 ⊢ (𝐴 ⊆ Sℋ → ∪ 𝐴 ⊆ ℋ) |
7 | spanss2 28505 | . 2 ⊢ (∪ 𝐴 ⊆ ℋ → ∪ 𝐴 ⊆ (span‘∪ 𝐴)) | |
8 | 6, 7 | syl 17 | 1 ⊢ (𝐴 ⊆ Sℋ → ∪ 𝐴 ⊆ (span‘∪ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3707 𝒫 cpw 4294 ∪ cuni 4580 ‘cfv 6041 ℋchil 28077 Sℋ csh 28086 spancspn 28090 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-rep 4915 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-cnex 10176 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-i2m1 10188 ax-1ne0 10189 ax-rrecex 10192 ax-cnre 10193 ax-hilex 28157 ax-hfvadd 28158 ax-hv0cl 28161 ax-hfvmul 28163 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-ral 3047 df-rex 3048 df-reu 3049 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-int 4620 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-om 7223 df-wrecs 7568 df-recs 7629 df-rdg 7667 df-map 8017 df-nn 11205 df-hlim 28130 df-sh 28365 df-ch 28379 df-span 28469 |
This theorem is referenced by: (None) |
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