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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 29015 | . . . . 5 ⊢ Sℋ ⊆ 𝒫 ℋ | |
2 | sstr 3973 | . . . . 5 ⊢ ((𝐴 ⊆ Sℋ ∧ Sℋ ⊆ 𝒫 ℋ) → 𝐴 ⊆ 𝒫 ℋ) | |
3 | 1, 2 | mpan2 689 | . . . 4 ⊢ (𝐴 ⊆ Sℋ → 𝐴 ⊆ 𝒫 ℋ) |
4 | 3 | unissd 4854 | . . 3 ⊢ (𝐴 ⊆ Sℋ → ∪ 𝐴 ⊆ ∪ 𝒫 ℋ) |
5 | unipw 5333 | . . 3 ⊢ ∪ 𝒫 ℋ = ℋ | |
6 | 4, 5 | sseqtrdi 4015 | . 2 ⊢ (𝐴 ⊆ Sℋ → ∪ 𝐴 ⊆ ℋ) |
7 | spanss2 29114 | . 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 3934 𝒫 cpw 4537 ∪ cuni 4830 ‘cfv 6348 ℋchba 28688 Sℋ csh 28697 spancspn 28701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-1cn 10587 ax-addcl 10589 ax-hilex 28768 ax-hfvadd 28769 ax-hv0cl 28772 ax-hfvmul 28774 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-map 8400 df-nn 11631 df-hlim 28741 df-sh 28976 df-ch 28990 df-span 29078 |
This theorem is referenced by: (None) |
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