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Mirrors > Home > MPE Home > Th. List > uncld | Structured version Visualization version Unicode version |
Description: The union of two closed sets is closed. Equivalent to Theorem 6.1(3) of [Munkres] p. 93. (Contributed by NM, 5-Oct-2006.) |
Ref | Expression |
---|---|
uncld |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difundi 3879 | . . 3 | |
2 | cldrcl 20830 | . . . . 5 | |
3 | 2 | adantr 481 | . . . 4 |
4 | eqid 2622 | . . . . . 6 | |
5 | 4 | cldopn 20835 | . . . . 5 |
6 | 5 | adantr 481 | . . . 4 |
7 | 4 | cldopn 20835 | . . . . 5 |
8 | 7 | adantl 482 | . . . 4 |
9 | inopn 20704 | . . . 4 | |
10 | 3, 6, 8, 9 | syl3anc 1326 | . . 3 |
11 | 1, 10 | syl5eqel 2705 | . 2 |
12 | 4 | cldss 20833 | . . . . 5 |
13 | 4 | cldss 20833 | . . . . 5 |
14 | 12, 13 | anim12i 590 | . . . 4 |
15 | unss 3787 | . . . 4 | |
16 | 14, 15 | sylib 208 | . . 3 |
17 | 4 | iscld2 20832 | . . 3 |
18 | 3, 16, 17 | syl2anc 693 | . 2 |
19 | 11, 18 | mpbird 247 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wcel 1990 cdif 3571 cun 3572 cin 3573 wss 3574 cuni 4436 cfv 5888 ctop 20698 ccld 20820 |
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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 df-top 20699 df-cld 20823 |
This theorem is referenced by: iscldtop 20899 paste 21098 lpcls 21168 dvasin 33496 dvacos 33497 dvreasin 33498 dvreacos 33499 |
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