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Mirrors > Home > ILE Home > Th. List > un0addcl | 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 2237 | . . . 4 |
3 | elun 3268 | . . . 4 | |
4 | 2, 3 | bitri 183 | . . 3 |
5 | 1 | eleq2i 2237 | . . . . . 6 |
6 | elun 3268 | . . . . . 6 | |
7 | 5, 6 | bitri 183 | . . . . 5 |
8 | ssun1 3290 | . . . . . . . . 9 | |
9 | 8, 1 | sseqtrri 3182 | . . . . . . . 8 |
10 | un0addcl.3 | . . . . . . . 8 | |
11 | 9, 10 | sselid 3145 | . . . . . . 7 |
12 | 11 | expr 373 | . . . . . 6 |
13 | un0addcl.1 | . . . . . . . . . . 11 | |
14 | 13 | sselda 3147 | . . . . . . . . . 10 |
15 | 14 | addid2d 8069 | . . . . . . . . 9 |
16 | 9 | a1i 9 | . . . . . . . . . 10 |
17 | 16 | sselda 3147 | . . . . . . . . 9 |
18 | 15, 17 | eqeltrd 2247 | . . . . . . . 8 |
19 | elsni 3601 | . . . . . . . . . 10 | |
20 | 19 | oveq1d 5868 | . . . . . . . . 9 |
21 | 20 | eleq1d 2239 | . . . . . . . 8 |
22 | 18, 21 | syl5ibrcom 156 | . . . . . . 7 |
23 | 22 | impancom 258 | . . . . . 6 |
24 | 12, 23 | jaodan 792 | . . . . 5 |
25 | 7, 24 | sylan2b 285 | . . . 4 |
26 | 0cnd 7913 | . . . . . . . . . . 11 | |
27 | 26 | snssd 3725 | . . . . . . . . . 10 |
28 | 13, 27 | unssd 3303 | . . . . . . . . 9 |
29 | 1, 28 | eqsstrid 3193 | . . . . . . . 8 |
30 | 29 | sselda 3147 | . . . . . . 7 |
31 | 30 | addid1d 8068 | . . . . . 6 |
32 | simpr 109 | . . . . . 6 | |
33 | 31, 32 | eqeltrd 2247 | . . . . 5 |
34 | elsni 3601 | . . . . . . 7 | |
35 | 34 | oveq2d 5869 | . . . . . 6 |
36 | 35 | eleq1d 2239 | . . . . 5 |
37 | 33, 36 | syl5ibrcom 156 | . . . 4 |
38 | 25, 37 | jaod 712 | . . 3 |
39 | 4, 38 | syl5bi 151 | . 2 |
40 | 39 | impr 377 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 703 wceq 1348 wcel 2141 cun 3119 wss 3121 csn 3583 (class class class)co 5853 cc 7772 cc0 7774 caddc 7777 |
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-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-1cn 7867 ax-icn 7869 ax-addcl 7870 ax-mulcl 7872 ax-addcom 7874 ax-i2m1 7879 ax-0id 7882 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-iota 5160 df-fv 5206 df-ov 5856 |
This theorem is referenced by: nn0addcl 9170 |
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