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Mirrors > Home > ILE Home > Th. List > iuncld | Unicode version |
Description: A finite indexed union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.) (Revised by Jim Kingdon, 10-Mar-2023.) |
Ref | Expression |
---|---|
iuncld.1 |
Ref | Expression |
---|---|
iuncld |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iuneq1 3821 | . . 3 | |
2 | 1 | eleq1d 2206 | . 2 |
3 | iuneq1 3821 | . . 3 | |
4 | 3 | eleq1d 2206 | . 2 |
5 | iuneq1 3821 | . . 3 | |
6 | 5 | eleq1d 2206 | . 2 |
7 | iuneq1 3821 | . . 3 | |
8 | 7 | eleq1d 2206 | . 2 |
9 | 0iun 3865 | . . . 4 | |
10 | 0cld 12270 | . . . 4 | |
11 | 9, 10 | eqeltrid 2224 | . . 3 |
12 | 11 | 3ad2ant1 1002 | . 2 |
13 | simpr 109 | . . . 4 | |
14 | nfcsb1v 3030 | . . . . . . . 8 | |
15 | csbeq1a 3007 | . . . . . . . 8 | |
16 | 14, 15 | iunxsngf 3885 | . . . . . . 7 |
17 | 16 | elv 2685 | . . . . . 6 |
18 | simprr 521 | . . . . . . . 8 | |
19 | 18 | eldifad 3077 | . . . . . . 7 |
20 | simpll3 1022 | . . . . . . 7 | |
21 | 14 | nfel1 2290 | . . . . . . . 8 |
22 | 15 | eleq1d 2206 | . . . . . . . 8 |
23 | 21, 22 | rspc 2778 | . . . . . . 7 |
24 | 19, 20, 23 | sylc 62 | . . . . . 6 |
25 | 17, 24 | eqeltrid 2224 | . . . . 5 |
26 | 25 | adantr 274 | . . . 4 |
27 | iunxun 3887 | . . . . 5 | |
28 | uncld 12271 | . . . . 5 | |
29 | 27, 28 | eqeltrid 2224 | . . . 4 |
30 | 13, 26, 29 | syl2anc 408 | . . 3 |
31 | 30 | ex 114 | . 2 |
32 | simp2 982 | . 2 | |
33 | 2, 4, 6, 8, 12, 31, 32 | findcard2sd 6779 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 wral 2414 cvv 2681 csb 2998 cdif 3063 cun 3064 wss 3066 c0 3358 csn 3522 cuni 3731 ciun 3808 cfv 5118 cfn 6627 ctop 12153 ccld 12250 |
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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-er 6422 df-en 6628 df-fin 6630 df-top 12154 df-cld 12253 |
This theorem is referenced by: unicld 12274 |
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