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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 | β’ (π΄ β Sβ β βͺ π΄ β (spanββͺ π΄)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | shsspwh 30486 | . . . . 5 β’ Sβ β π« β | |
| 2 | sstr 3989 | . . . . 5 β’ ((π΄ β Sβ β§ Sβ β π« β) β π΄ β π« β) | |
| 3 | 1, 2 | mpan2 689 | . . . 4 β’ (π΄ β Sβ β π΄ β π« β) |
| 4 | 3 | unissd 4917 | . . 3 β’ (π΄ β Sβ β βͺ π΄ β βͺ π« β) |
| 5 | unipw 5449 | . . 3 β’ βͺ π« β = β | |
| 6 | 4, 5 | sseqtrdi 4031 | . 2 β’ (π΄ β Sβ β βͺ π΄ β β) |
| 7 | spanss2 30585 | . 2 β’ (βͺ π΄ β β β βͺ π΄ β (spanββͺ π΄)) | |
| 8 | 6, 7 | syl 17 | 1 β’ (π΄ β Sβ β βͺ π΄ β (spanββͺ π΄)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wss 3947 π« cpw 4601 βͺ cuni 4907 βcfv 6540 βchba 30159 Sβ csh 30168 spancspn 30172 |
| 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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-1cn 11164 ax-addcl 11166 ax-hilex 30239 ax-hfvadd 30240 ax-hv0cl 30243 ax-hfvmul 30245 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-map 8818 df-nn 12209 df-hlim 30212 df-sh 30447 df-ch 30461 df-span 30549 |
| This theorem is referenced by: (None) |
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