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Mirrors > Home > MPE Home > Th. List > uzin | Structured version Visualization version Unicode version |
Description: Intersection of two upper intervals of integers. (Contributed by Mario Carneiro, 24-Dec-2013.) |
Ref | Expression |
---|---|
uzin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uztric 11709 | . 2 | |
2 | uzss 11708 | . . . . 5 | |
3 | sseqin2 3817 | . . . . 5 | |
4 | 2, 3 | sylib 208 | . . . 4 |
5 | eluzle 11700 | . . . . . 6 | |
6 | iftrue 4092 | . . . . . 6 | |
7 | 5, 6 | syl 17 | . . . . 5 |
8 | 7 | fveq2d 6195 | . . . 4 |
9 | 4, 8 | eqtr4d 2659 | . . 3 |
10 | uzss 11708 | . . . . 5 | |
11 | df-ss 3588 | . . . . 5 | |
12 | 10, 11 | sylib 208 | . . . 4 |
13 | eluzel2 11692 | . . . . . . . . . . 11 | |
14 | eluzelz 11697 | . . . . . . . . . . 11 | |
15 | zre 11381 | . . . . . . . . . . . 12 | |
16 | zre 11381 | . . . . . . . . . . . 12 | |
17 | letri3 10123 | . . . . . . . . . . . 12 | |
18 | 15, 16, 17 | syl2an 494 | . . . . . . . . . . 11 |
19 | 13, 14, 18 | syl2anc 693 | . . . . . . . . . 10 |
20 | eluzle 11700 | . . . . . . . . . . 11 | |
21 | 20 | biantrurd 529 | . . . . . . . . . 10 |
22 | 19, 21 | bitr4d 271 | . . . . . . . . 9 |
23 | 22 | biimprcd 240 | . . . . . . . 8 |
24 | 6 | eqeq1d 2624 | . . . . . . . 8 |
25 | 23, 24 | sylibrd 249 | . . . . . . 7 |
26 | 25 | com12 32 | . . . . . 6 |
27 | iffalse 4095 | . . . . . 6 | |
28 | 26, 27 | pm2.61d1 171 | . . . . 5 |
29 | 28 | fveq2d 6195 | . . . 4 |
30 | 12, 29 | eqtr4d 2659 | . . 3 |
31 | 9, 30 | jaoi 394 | . 2 |
32 | 1, 31 | syl 17 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wo 383 wa 384 wceq 1483 wcel 1990 cin 3573 wss 3574 cif 4086 class class class wbr 4653 cfv 5888 cr 9935 cle 10075 cz 11377 cuz 11687 |
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-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 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-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 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-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-neg 10269 df-z 11378 df-uz 11688 |
This theorem is referenced by: uzin2 14084 explecnv 14597 uzrest 21701 |
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