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Mirrors > Home > MPE Home > Th. List > ipopos | Structured version Visualization version Unicode version |
Description: The inclusion poset on a family of sets is actually a poset. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
Ref | Expression |
---|---|
ipopos.i | toInc |
Ref | Expression |
---|---|
ipopos |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ipopos.i | . . . . 5 toInc | |
2 | fvex 6201 | . . . . 5 toInc | |
3 | 1, 2 | eqeltri 2697 | . . . 4 |
4 | 3 | a1i 11 | . . 3 |
5 | 1 | ipobas 17155 | . . 3 |
6 | eqidd 2623 | . . 3 | |
7 | ssid 3624 | . . . 4 | |
8 | eqid 2622 | . . . . . 6 | |
9 | 1, 8 | ipole 17158 | . . . . 5 |
10 | 9 | 3anidm23 1385 | . . . 4 |
11 | 7, 10 | mpbiri 248 | . . 3 |
12 | 1, 8 | ipole 17158 | . . . . 5 |
13 | 1, 8 | ipole 17158 | . . . . . 6 |
14 | 13 | 3com23 1271 | . . . . 5 |
15 | 12, 14 | anbi12d 747 | . . . 4 |
16 | simpl 473 | . . . . 5 | |
17 | simpr 477 | . . . . 5 | |
18 | 16, 17 | eqssd 3620 | . . . 4 |
19 | 15, 18 | syl6bi 243 | . . 3 |
20 | sstr 3611 | . . . . 5 | |
21 | 20 | a1i 11 | . . . 4 |
22 | 12 | 3adant3r3 1276 | . . . . 5 |
23 | 1, 8 | ipole 17158 | . . . . . 6 |
24 | 23 | 3adant3r1 1274 | . . . . 5 |
25 | 22, 24 | anbi12d 747 | . . . 4 |
26 | 1, 8 | ipole 17158 | . . . . 5 |
27 | 26 | 3adant3r2 1275 | . . . 4 |
28 | 21, 25, 27 | 3imtr4d 283 | . . 3 |
29 | 4, 5, 6, 11, 19, 28 | isposd 16955 | . 2 |
30 | fvprc 6185 | . . . 4 toInc | |
31 | 1, 30 | syl5eq 2668 | . . 3 |
32 | 0pos 16954 | . . 3 | |
33 | 31, 32 | syl6eqel 2709 | . 2 |
34 | 29, 33 | pm2.61i 176 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 cvv 3200 wss 3574 c0 3915 class class class wbr 4653 cfv 5888 cple 15948 cpo 16940 toInccipo 17151 |
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-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 |
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-reu 2919 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-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-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-tset 15960 df-ple 15961 df-ocomp 15963 df-poset 16946 df-ipo 17152 |
This theorem is referenced by: isipodrs 17161 mrelatglb 17184 mrelatglb0 17185 mrelatlub 17186 mreclatBAD 17187 |
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