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Mirrors > Home > MPE Home > Th. List > un0addcl | Structured version Visualization version Unicode version |
Description: If is closed under addition, then so is . (Contributed by Mario Carneiro, 17-Jul-2014.) |
Ref | Expression |
---|---|
un0addcl.1 | |
un0addcl.2 | |
un0addcl.3 |
Ref | Expression |
---|---|
un0addcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | un0addcl.2 | . . . . 5 | |
2 | 1 | eleq2i 2693 | . . . 4 |
3 | elun 3753 | . . . 4 | |
4 | 2, 3 | bitri 264 | . . 3 |
5 | 1 | eleq2i 2693 | . . . . . 6 |
6 | elun 3753 | . . . . . 6 | |
7 | 5, 6 | bitri 264 | . . . . 5 |
8 | ssun1 3776 | . . . . . . . . 9 | |
9 | 8, 1 | sseqtr4i 3638 | . . . . . . . 8 |
10 | un0addcl.3 | . . . . . . . 8 | |
11 | 9, 10 | sseldi 3601 | . . . . . . 7 |
12 | 11 | expr 643 | . . . . . 6 |
13 | un0addcl.1 | . . . . . . . . . . 11 | |
14 | 13 | sselda 3603 | . . . . . . . . . 10 |
15 | 14 | addid2d 10237 | . . . . . . . . 9 |
16 | 9 | a1i 11 | . . . . . . . . . 10 |
17 | 16 | sselda 3603 | . . . . . . . . 9 |
18 | 15, 17 | eqeltrd 2701 | . . . . . . . 8 |
19 | elsni 4194 | . . . . . . . . . 10 | |
20 | 19 | oveq1d 6665 | . . . . . . . . 9 |
21 | 20 | eleq1d 2686 | . . . . . . . 8 |
22 | 18, 21 | syl5ibrcom 237 | . . . . . . 7 |
23 | 22 | impancom 456 | . . . . . 6 |
24 | 12, 23 | jaodan 826 | . . . . 5 |
25 | 7, 24 | sylan2b 492 | . . . 4 |
26 | 0cnd 10033 | . . . . . . . . . . 11 | |
27 | 26 | snssd 4340 | . . . . . . . . . 10 |
28 | 13, 27 | unssd 3789 | . . . . . . . . 9 |
29 | 1, 28 | syl5eqss 3649 | . . . . . . . 8 |
30 | 29 | sselda 3603 | . . . . . . 7 |
31 | 30 | addid1d 10236 | . . . . . 6 |
32 | simpr 477 | . . . . . 6 | |
33 | 31, 32 | eqeltrd 2701 | . . . . 5 |
34 | elsni 4194 | . . . . . . 7 | |
35 | 34 | oveq2d 6666 | . . . . . 6 |
36 | 35 | eleq1d 2686 | . . . . 5 |
37 | 33, 36 | syl5ibrcom 237 | . . . 4 |
38 | 25, 37 | jaod 395 | . . 3 |
39 | 4, 38 | syl5bi 232 | . 2 |
40 | 39 | impr 649 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wo 383 wa 384 wceq 1483 wcel 1990 cun 3572 wss 3574 csn 4177 (class class class)co 6650 cc 9934 cc0 9936 caddc 9939 |
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 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 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-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 |
This theorem is referenced by: nn0addcl 11328 plyaddlem 23971 plymullem 23972 |
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