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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 3826 | . . 3 | |
2 | 1 | eleq1d 2208 | . 2 |
3 | iuneq1 3826 | . . 3 | |
4 | 3 | eleq1d 2208 | . 2 |
5 | iuneq1 3826 | . . 3 | |
6 | 5 | eleq1d 2208 | . 2 |
7 | iuneq1 3826 | . . 3 | |
8 | 7 | eleq1d 2208 | . 2 |
9 | 0iun 3870 | . . . 4 | |
10 | 0cld 12284 | . . . 4 | |
11 | 9, 10 | eqeltrid 2226 | . . 3 |
12 | 11 | 3ad2ant1 1002 | . 2 |
13 | simpr 109 | . . . 4 | |
14 | nfcsb1v 3035 | . . . . . . . 8 | |
15 | csbeq1a 3012 | . . . . . . . 8 | |
16 | 14, 15 | iunxsngf 3890 | . . . . . . 7 |
17 | 16 | elv 2690 | . . . . . 6 |
18 | simprr 521 | . . . . . . . 8 | |
19 | 18 | eldifad 3082 | . . . . . . 7 |
20 | simpll3 1022 | . . . . . . 7 | |
21 | 14 | nfel1 2292 | . . . . . . . 8 |
22 | 15 | eleq1d 2208 | . . . . . . . 8 |
23 | 21, 22 | rspc 2783 | . . . . . . 7 |
24 | 19, 20, 23 | sylc 62 | . . . . . 6 |
25 | 17, 24 | eqeltrid 2226 | . . . . 5 |
26 | 25 | adantr 274 | . . . 4 |
27 | iunxun 3892 | . . . . 5 | |
28 | uncld 12285 | . . . . 5 | |
29 | 27, 28 | eqeltrid 2226 | . . . 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 6786 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 wral 2416 cvv 2686 csb 3003 cdif 3068 cun 3069 wss 3071 c0 3363 csn 3527 cuni 3736 ciun 3813 cfv 5123 cfn 6634 ctop 12167 ccld 12264 |
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-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
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 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-reu 2423 df-rab 2425 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-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 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-er 6429 df-en 6635 df-fin 6637 df-top 12168 df-cld 12267 |
This theorem is referenced by: unicld 12288 |
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