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Mirrors > Home > ILE Home > Th. List > fival | Unicode version |
Description: The set of all the finite intersections of the elements of . (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.) |
Ref | Expression |
---|---|
fival |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fi 6915 | . 2 | |
2 | pweq 3547 | . . . . 5 | |
3 | 2 | ineq1d 3308 | . . . 4 |
4 | 3 | rexeqdv 2659 | . . 3 |
5 | 4 | abbidv 2275 | . 2 |
6 | elex 2723 | . 2 | |
7 | simpr 109 | . . . . . . 7 | |
8 | elinel1 3294 | . . . . . . . . 9 | |
9 | 8 | elpwid 3555 | . . . . . . . 8 |
10 | eqvisset 2722 | . . . . . . . . . . . 12 | |
11 | intexr 4113 | . . . . . . . . . . . 12 | |
12 | 10, 11 | syl 14 | . . . . . . . . . . 11 |
13 | 12 | adantl 275 | . . . . . . . . . 10 |
14 | 13 | neneqd 2348 | . . . . . . . . 9 |
15 | elinel2 3295 | . . . . . . . . . . 11 | |
16 | 15 | adantr 274 | . . . . . . . . . 10 |
17 | fin0or 6833 | . . . . . . . . . . 11 | |
18 | 17 | orcomd 719 | . . . . . . . . . 10 |
19 | 16, 18 | syl 14 | . . . . . . . . 9 |
20 | 14, 19 | ecased 1331 | . . . . . . . 8 |
21 | intssuni2m 3833 | . . . . . . . 8 | |
22 | 9, 20, 21 | syl2an2r 585 | . . . . . . 7 |
23 | 7, 22 | eqsstrd 3164 | . . . . . 6 |
24 | velpw 3551 | . . . . . 6 | |
25 | 23, 24 | sylibr 133 | . . . . 5 |
26 | 25 | rexlimiva 2569 | . . . 4 |
27 | 26 | abssi 3203 | . . 3 |
28 | uniexg 4401 | . . . 4 | |
29 | 28 | pwexd 4144 | . . 3 |
30 | ssexg 4105 | . . 3 | |
31 | 27, 29, 30 | sylancr 411 | . 2 |
32 | 1, 5, 6, 31 | fvmptd3 5563 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 698 wceq 1335 wex 1472 wcel 2128 cab 2143 wne 2327 wrex 2436 cvv 2712 cin 3101 wss 3102 c0 3395 cpw 3544 cuni 3774 cint 3809 cfv 5172 cfn 6687 cfi 6914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-iinf 4549 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-br 3968 df-opab 4028 df-mpt 4029 df-id 4255 df-suc 4333 df-iom 4552 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-er 6482 df-en 6688 df-fin 6690 df-fi 6915 |
This theorem is referenced by: elfi 6917 fi0 6921 |
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