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Mirrors > Home > HSE Home > Th. List > shunssi | Structured version Unicode version |
Description: Union is smaller than subspace sum. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shincl.1 |
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shincl.2 |
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Ref | Expression |
---|---|
shunssi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shincl.1 |
. . . . . . 7
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2 | 1 | sheli 24761 |
. . . . . 6
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3 | ax-hvaddid 24551 |
. . . . . . 7
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4 | 3 | eqcomd 2459 |
. . . . . 6
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5 | 2, 4 | syl 16 |
. . . . 5
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6 | shincl.2 |
. . . . . . 7
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7 | sh0 24763 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | 6, 7 | ax-mp 5 |
. . . . . 6
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9 | rspceov 6230 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | 8, 9 | mp3an2 1303 |
. . . . 5
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11 | 5, 10 | mpdan 668 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 6 | sheli 24761 |
. . . . . 6
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13 | hvaddid2 24570 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 13 | eqcomd 2459 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 12, 14 | syl 16 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | sh0 24763 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 1, 16 | ax-mp 5 |
. . . . . 6
![]() ![]() ![]() ![]() |
18 | rspceov 6230 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 17, 18 | mp3an1 1302 |
. . . . 5
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20 | 15, 19 | mpdan 668 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 11, 20 | jaoi 379 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | elun 3598 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | 1, 6 | shseli 24864 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 21, 22, 23 | 3imtr4i 266 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | ssriv 3461 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-rep 4504 ax-sep 4514 ax-nul 4522 ax-pow 4571 ax-pr 4632 ax-un 6475 ax-resscn 9443 ax-1cn 9444 ax-icn 9445 ax-addcl 9446 ax-addrcl 9447 ax-mulcl 9448 ax-mulrcl 9449 ax-mulcom 9450 ax-addass 9451 ax-mulass 9452 ax-distr 9453 ax-i2m1 9454 ax-1ne0 9455 ax-1rid 9456 ax-rnegex 9457 ax-rrecex 9458 ax-cnre 9459 ax-pre-lttri 9460 ax-pre-lttrn 9461 ax-pre-ltadd 9462 ax-hilex 24546 ax-hfvadd 24547 ax-hvcom 24548 ax-hvass 24549 ax-hv0cl 24550 ax-hvaddid 24551 ax-hfvmul 24552 ax-hvmulid 24553 ax-hvdistr2 24556 ax-hvmul0 24557 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-nel 2647 df-ral 2800 df-rex 2801 df-reu 2802 df-rab 2804 df-v 3073 df-sbc 3288 df-csb 3390 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-nul 3739 df-if 3893 df-pw 3963 df-sn 3979 df-pr 3981 df-op 3985 df-uni 4193 df-iun 4274 df-br 4394 df-opab 4452 df-mpt 4453 df-id 4737 df-po 4742 df-so 4743 df-xp 4947 df-rel 4948 df-cnv 4949 df-co 4950 df-dm 4951 df-rn 4952 df-res 4953 df-ima 4954 df-iota 5482 df-fun 5521 df-fn 5522 df-f 5523 df-f1 5524 df-fo 5525 df-f1o 5526 df-fv 5527 df-riota 6154 df-ov 6196 df-oprab 6197 df-mpt2 6198 df-er 7204 df-en 7414 df-dom 7415 df-sdom 7416 df-pnf 9524 df-mnf 9525 df-ltxr 9527 df-sub 9701 df-neg 9702 df-grpo 23823 df-ablo 23914 df-hvsub 24518 df-sh 24754 df-shs 24856 |
This theorem is referenced by: shsval2i 24935 shjshsi 25040 spanuni 25092 |
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