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Mirrors > Home > ILE Home > Th. List > fiuni | Unicode version |
Description: The union of the finite intersections of a set is simply the union of the set itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.) |
Ref | Expression |
---|---|
fiuni |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssfii 6862 | . . 3 | |
2 | 1 | unissd 3760 | . 2 |
3 | eluni 3739 | . . . . 5 | |
4 | 3 | biimpi 119 | . . . 4 |
5 | 4 | adantl 275 | . . 3 |
6 | simprr 521 | . . . . 5 | |
7 | elfi2 6860 | . . . . . 6 | |
8 | 7 | ad2antrr 479 | . . . . 5 |
9 | 6, 8 | mpbid 146 | . . . 4 |
10 | simprr 521 | . . . . . 6 | |
11 | eldifi 3198 | . . . . . . . . . 10 | |
12 | 11 | elin1d 3265 | . . . . . . . . 9 |
13 | 12 | elpwid 3521 | . . . . . . . 8 |
14 | 13 | ad2antrl 481 | . . . . . . 7 |
15 | eldifsni 3652 | . . . . . . . . 9 | |
16 | 11 | elin2d 3266 | . . . . . . . . . 10 |
17 | fin0 6779 | . . . . . . . . . 10 | |
18 | 16, 17 | syl 14 | . . . . . . . . 9 |
19 | 15, 18 | mpbid 146 | . . . . . . . 8 |
20 | 19 | ad2antrl 481 | . . . . . . 7 |
21 | intssuni2m 3795 | . . . . . . 7 | |
22 | 14, 20, 21 | syl2anc 408 | . . . . . 6 |
23 | 10, 22 | eqsstrd 3133 | . . . . 5 |
24 | simplrl 524 | . . . . 5 | |
25 | 23, 24 | sseldd 3098 | . . . 4 |
26 | 9, 25 | rexlimddv 2554 | . . 3 |
27 | 5, 26 | exlimddv 1870 | . 2 |
28 | 2, 27 | eqelssd 3116 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 wne 2308 wrex 2417 cdif 3068 cin 3070 wss 3071 c0 3363 cpw 3510 csn 3527 cuni 3736 cint 3771 cfv 5123 cfn 6634 cfi 6856 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1o 6313 df-er 6429 df-en 6635 df-fin 6637 df-fi 6857 |
This theorem is referenced by: fipwssg 6867 |
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