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| Mirrors > Home > MPE Home > Th. List > uniiccdif | Structured version Visualization version Unicode version | ||
| Description: A union of closed intervals differs from the equivalent union of open intervals by a nullset. (Contributed by Mario Carneiro, 25-Mar-2015.) |
| Ref | Expression |
|---|---|
| uniioombl.1 |
             |
| Ref | Expression |
|---|---|
| uniiccdif |
     ![[,]](_icc.gif "[,]"){kind=small} ![/\](wedge.gif "/\"){kind=small}    ![\](setminus.gif "\"){kind=small}
  |
is distinct from all other
variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssun1 3776 |
. . 3
    | |
| 2 | uniioombl.1 |
. . . . . . . 8
| |
| 3 | ovolfcl 23235 |
. . . . . . . 8
 | |
| 4 | 2, 3 | sylan 488 |
. . . . . . 7
|
| 5 | rexr 10085 |
. . . . . . . 8
 | |
| 6 | rexr 10085 |
. . . . . . . 8
| |
| 7 | id 22 |
. . . . . . . 8
| |
| 8 | prunioo 12301 |
. . . . . . . 8
   | |
| 9 | 5, 6, 7, 8 | syl3an 1368 |
. . . . . . 7
|
| 10 | 4, 9 | syl 17 |
. . . . . 6
|
| 11 | fvco3 6275 |
. . . . . . . . 9
| |
| 12 | 2, 11 | sylan 488 |
. . . . . . . 8
|
| 13 | inss2 3834 |
. . . . . . . . . . . 12
| |
| 14 | 2 | ffvelrnda 6359 |
. . . . . . . . . . . 12
|
| 15 | 13, 14 | sseldi 3601 |
. . . . . . . . . . 11
|
| 16 | 1st2nd2 7205 |
. . . . . . . . . . 11
  | |
| 17 | 15, 16 | syl 17 |
. . . . . . . . . 10
|
| 18 | 17 | fveq2d 6195 |
. . . . . . . . 9
|
| 19 | df-ov 6653 |
. . . . . . . . 9
| |
| 20 | 18, 19 | syl6eqr 2674 |
. . . . . . . 8
|
| 21 | 12, 20 | eqtrd 2656 |
. . . . . . 7
|
| 22 | df-pr 4180 |
. . . . . . . 8
| |
| 23 | fvco3 6275 |
. . . . . . . . . 10
| |
| 24 | 2, 23 | sylan 488 |
. . . . . . . . 9
|
| 25 | fvco3 6275 |
. . . . . . . . . 10
| |
| 26 | 2, 25 | sylan 488 |
. . . . . . . . 9
|
| 27 | 24, 26 | preq12d 4276 |
. . . . . . . 8
|
| 28 | 22, 27 | syl5eqr 2670 |
. . . . . . 7
|
| 29 | 21, 28 | uneq12d 3768 |
. . . . . 6
|
| 30 | fvco3 6275 |
. . . . . . . 8
| |
| 31 | 2, 30 | sylan 488 |
. . . . . . 7
|
| 32 | 17 | fveq2d 6195 |
. . . . . . . 8
|
| 33 | df-ov 6653 |
. . . . . . . 8
| |
| 34 | 32, 33 | syl6eqr 2674 |
. . . . . . 7
|
| 35 | 31, 34 | eqtrd 2656 |
. . . . . 6
|
| 36 | 10, 29, 35 | 3eqtr4rd 2667 |
. . . . 5
|
| 37 | 36 | iuneq2dv 4542 |
. . . 4
 |
| 38 | iccf 12272 |
. . . . . . 7
 | |
| 39 | ffn 6045 |
. . . . . . 7
 | |
| 40 | 38, 39 | ax-mp 5 |
. . . . . 6
|
| 41 | rexpssxrxp 10084 |
. . . . . . . 8
| |
| 42 | 13, 41 | sstri 3612 |
. . . . . . 7
|
| 43 | fss 6056 |
. . . . . . 7
| |
| 44 | 2, 42, 43 | sylancl 694 |
. . . . . 6
|
| 45 | fnfco 6069 |
. . . . . 6
| |
| 46 | 40, 44, 45 | sylancr 695 |
. . . . 5
|
| 47 | fniunfv 6505 |
. . . . 5
| |
| 48 | 46, 47 | syl 17 |
. . . 4
|
| 49 | iunun 4604 |
. . . . 5
| |
| 50 | ioof 12271 |
. . . . . . . . 9
| |
| 51 | ffn 6045 |
. . . . . . . . 9
| |
| 52 | 50, 51 | ax-mp 5 |
. . . . . . . 8
|
| 53 | fnfco 6069 |
. . . . . . . 8
| |
| 54 | 52, 44, 53 | sylancr 695 |
. . . . . . 7
|
| 55 | fniunfv 6505 |
. . . . . . 7
| |
| 56 | 54, 55 | syl 17 |
. . . . . 6
|
| 57 | iunun 4604 |
. . . . . . 7
| |
| 58 | fo1st 7188 |
. . . . . . . . . . . . . 14
  | |
| 59 | fofn 6117 |
. . . . . . . . . . . . . 14
| |
| 60 | 58, 59 | ax-mp 5 |
. . . . . . . . . . . . 13
|
| 61 | ssv 3625 |
. . . . . . . . . . . . . 14
| |
| 62 | fss 6056 |
. . . . . . . . . . . . . 14
| |
| 63 | 2, 61, 62 | sylancl 694 |
. . . . . . . . . . . . 13
|
| 64 | fnfco 6069 |
. . . . . . . . . . . . 13
| |
| 65 | 60, 63, 64 | sylancr 695 |
. . . . . . . . . . . 12
|
| 66 | fnfun 5988 |
. . . . . . . . . . . 12
 | |
| 67 | 65, 66 | syl 17 |
. . . . . . . . . . 11
|
| 68 | fndm 5990 |
. . . . . . . . . . . 12
 | |
| 69 | eqimss2 3658 |
. . . . . . . . . . . 12
| |
| 70 | 65, 68, 69 | 3syl 18 |
. . . . . . . . . . 11
|
| 71 | dfimafn2 6246 |
. . . . . . . . . . 11
| |
| 72 | 67, 70, 71 | syl2anc 693 |
. . . . . . . . . 10
|
| 73 | fnima 6010 |
. . . . . . . . . . 11
| |
| 74 | 65, 73 | syl 17 |
. . . . . . . . . 10
|
| 75 | 72, 74 | eqtr3d 2658 |
. . . . . . . . 9
|
| 76 | rnco2 5642 |
. . . . . . . . 9
| |
| 77 | 75, 76 | syl6eq 2672 |
. . . . . . . 8
|
| 78 | fo2nd 7189 |
. . . . . . . . . . . . . 14
| |
| 79 | fofn 6117 |
. . . . . . . . . . . . . 14
| |
| 80 | 78, 79 | ax-mp 5 |
. . . . . . . . . . . . 13
|
| 81 | fnfco 6069 |
. . . . . . . . . . . . 13
| |
| 82 | 80, 63, 81 | sylancr 695 |
. . . . . . . . . . . 12
|
| 83 | fnfun 5988 |
. . . . . . . . . . . 12
| |
| 84 | 82, 83 | syl 17 |
. . . . . . . . . . 11
|
| 85 | fndm 5990 |
. . . . . . . . . . . 12
| |
| 86 | eqimss2 3658 |
. . . . . . . . . . . 12
| |
| 87 | 82, 85, 86 | 3syl 18 |
. . . . . . . . . . 11
|
| 88 | dfimafn2 6246 |
. . . . . . . . . . 11
| |
| 89 | 84, 87, 88 | syl2anc 693 |
. . . . . . . . . 10
|
| 90 | fnima 6010 |
. . . . . . . . . . 11
| |
| 91 | 82, 90 | syl 17 |
. . . . . . . . . 10
|
| 92 | 89, 91 | eqtr3d 2658 |
. . . . . . . . 9
|
| 93 | rnco2 5642 |
. . . . . . . . 9
| |
| 94 | 92, 93 | syl6eq 2672 |
. . . . . . . 8
|
| 95 | 77, 94 | uneq12d 3768 |
. . . . . . 7
|
| 96 | 57, 95 | syl5eq 2668 |
. . . . . 6
|
| 97 | 56, 96 | uneq12d 3768 |
. . . . 5
|
| 98 | 49, 97 | syl5eq 2668 |
. . . 4
|
| 99 | 37, 48, 98 | 3eqtr3d 2664 |
. . 3
|
| 100 | 1, 99 | syl5sseqr 3654 |
. 2
|
| 101 | ovolficcss 23238 |
. . . . 5
| |
| 102 | 2, 101 | syl 17 |
. . . 4
|
| 103 | 102 | ssdifssd 3748 |
. . 3
|
| 104 | omelon 8543 |
. . . . . . . . . . 11
  | |
| 105 | nnenom 12779 |
. . . . . . . . . . . 12
 | |
| 106 | 105 | ensymi 8006 |
. . . . . . . . . . 11
|
| 107 | isnumi 8772 |
. . . . . . . . . . 11
 | |
| 108 | 104, 106, 107 | mp2an 708 |
. . . . . . . . . 10
|
| 109 | fofun 6116 |
. . . . . . . . . . . . 13
| |
| 110 | 58, 109 | ax-mp 5 |
. . . . . . . . . . . 12
|
| 111 | ssv 3625 |
. . . . . . . . . . . . 13
| |
| 112 | fof 6115 |
. . . . . . . . . . . . . . 15
| |
| 113 | 58, 112 | ax-mp 5 |
. . . . . . . . . . . . . 14
|
| 114 | 113 | fdmi 6052 |
. . . . . . . . . . . . 13
|
| 115 | 111, 114 | sseqtr4i 3638 |
. . . . . . . . . . . 12
|
| 116 | fores 6124 |
. . . . . . . . . . . 12
 | |
| 117 | 110, 115, 116 | mp2an 708 |
. . . . . . . . . . 11
|
| 118 | ffn 6045 |
. . . . . . . . . . . . 13
| |
| 119 | 2, 118 | syl 17 |
. . . . . . . . . . . 12
|
| 120 | dffn4 6121 |
. . . . . . . . . . . 12
| |
| 121 | 119, 120 | sylib 208 |
. . . . . . . . . . 11
|
| 122 | foco 6125 |
. . . . . . . . . . 11
| |
| 123 | 117, 121, 122 | sylancr 695 |
. . . . . . . . . 10
|
| 124 | fodomnum 8880 |
. . . . . . . . . 10
 | |
| 125 | 108, 123, 124 | mpsyl 68 |
. . . . . . . . 9
|
| 126 | domentr 8015 |
. . . . . . . . 9
| |
| 127 | 125, 105, 126 | sylancl 694 |
. . . . . . . 8
|
| 128 | fofun 6116 |
. . . . . . . . . . . . 13
| |
| 129 | 78, 128 | ax-mp 5 |
. . . . . . . . . . . 12
|
| 130 | fof 6115 |
. . . . . . . . . . . . . . 15
| |
| 131 | 78, 130 | ax-mp 5 |
. . . . . . . . . . . . . 14
|
| 132 | 131 | fdmi 6052 |
. . . . . . . . . . . . 13
|
| 133 | 111, 132 | sseqtr4i 3638 |
. . . . . . . . . . . 12
|
| 134 | fores 6124 |
. . . . . . . . . . . 12
| |
| 135 | 129, 133, 134 | mp2an 708 |
. . . . . . . . . . 11
|
| 136 | foco 6125 |
. . . . . . . . . . 11
| |
| 137 | 135, 121, 136 | sylancr 695 |
. . . . . . . . . 10
|
| 138 | fodomnum 8880 |
. . . . . . . . . 10
| |
| 139 | 108, 137, 138 | mpsyl 68 |
. . . . . . . . 9
|
| 140 | domentr 8015 |
. . . . . . . . 9
| |
| 141 | 139, 105, 140 | sylancl 694 |
. . . . . . . 8
|
| 142 | unctb 9027 |
. . . . . . . 8
| |
| 143 | 127, 141, 142 | syl2anc 693 |
. . . . . . 7
|
| 144 | reldom 7961 |
. . . . . . . 8
 | |
| 145 | 144 | brrelexi 5158 |
. . . . . . 7
|
| 146 | 143, 145 | syl 17 |
. . . . . 6
|
| 147 | ssid 3624 |
. . . . . . . 8
| |
| 148 | 147, 99 | syl5sseq 3653 |
. . . . . . 7
|
| 149 | ssundif 4052 |
. . . . . . 7
| |
| 150 | 148, 149 | sylib 208 |
. . . . . 6
|
| 151 | ssdomg 8001 |
. . . . . 6
| |
| 152 | 146, 150, 151 | sylc 65 |
. . . . 5
|
| 153 | domtr 8009 |
. . . . 5
| |
| 154 | 152, 143, 153 | syl2anc 693 |
. . . 4
|
| 155 | domentr 8015 |
. . . 4
| |
| 156 | 154, 106, 155 | sylancl 694 |
. . 3
|
| 157 | ovolctb2 23260 |
. . 3
| |
| 158 | 103, 156, 157 | syl2anc 693 |
. 2
|
| 159 | 100, 158 | jca 554 |
1
|
| Colors of variables: wff setvar class |
| Syntax hints: wi 4
wa 384
w3a 1037
wceq 1483
wcel 1990
cvv 3200
cdif 3571
cun 3572
cin 3573
wss 3574
cpw 4158
csn 4177
cpr 4179
cop 4183
cuni 4436
ciun 4520
class class class wbr 4653 cxp 5112 cdm 5114 crn 5115 cres 5116 cima 5117 ccom 5118 con0 5723 wfun 5882 wfn 5883 wf 5884 wfo 5886
cfv 5888
(class class class)co 6650 com 7065 c1st 7166 c2nd 7167 cen 7952 cdom 7953 ccrd 8761 cr 9935 cc0 9936 cxr 10073 cle 10075 cn 11020 cioo 12175 cicc 12178 covol 23231 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
| This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xadd 11947 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-xmet 19739 df-met 19740 df-ovol 23233 |
| This theorem is referenced by: uniioombllem3 23353 uniioombllem4 23354 uniioombllem5 23355 uniiccmbl 23358 |
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