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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 2204 | . . . 4 |
3 | elun 3212 | . . . 4 | |
4 | 2, 3 | bitri 183 | . . 3 |
5 | 1 | eleq2i 2204 | . . . . . 6 |
6 | elun 3212 | . . . . . 6 | |
7 | 5, 6 | bitri 183 | . . . . 5 |
8 | ssun1 3234 | . . . . . . . . 9 | |
9 | 8, 1 | sseqtrri 3127 | . . . . . . . 8 |
10 | un0addcl.3 | . . . . . . . 8 | |
11 | 9, 10 | sseldi 3090 | . . . . . . 7 |
12 | 11 | expr 372 | . . . . . 6 |
13 | un0addcl.1 | . . . . . . . . . . 11 | |
14 | 13 | sselda 3092 | . . . . . . . . . 10 |
15 | 14 | addid2d 7905 | . . . . . . . . 9 |
16 | 9 | a1i 9 | . . . . . . . . . 10 |
17 | 16 | sselda 3092 | . . . . . . . . 9 |
18 | 15, 17 | eqeltrd 2214 | . . . . . . . 8 |
19 | elsni 3540 | . . . . . . . . . 10 | |
20 | 19 | oveq1d 5782 | . . . . . . . . 9 |
21 | 20 | eleq1d 2206 | . . . . . . . 8 |
22 | 18, 21 | syl5ibrcom 156 | . . . . . . 7 |
23 | 22 | impancom 258 | . . . . . 6 |
24 | 12, 23 | jaodan 786 | . . . . 5 |
25 | 7, 24 | sylan2b 285 | . . . 4 |
26 | 0cnd 7752 | . . . . . . . . . . 11 | |
27 | 26 | snssd 3660 | . . . . . . . . . 10 |
28 | 13, 27 | unssd 3247 | . . . . . . . . 9 |
29 | 1, 28 | eqsstrid 3138 | . . . . . . . 8 |
30 | 29 | sselda 3092 | . . . . . . 7 |
31 | 30 | addid1d 7904 | . . . . . 6 |
32 | simpr 109 | . . . . . 6 | |
33 | 31, 32 | eqeltrd 2214 | . . . . 5 |
34 | elsni 3540 | . . . . . . 7 | |
35 | 34 | oveq2d 5783 | . . . . . 6 |
36 | 35 | eleq1d 2206 | . . . . 5 |
37 | 33, 36 | syl5ibrcom 156 | . . . 4 |
38 | 25, 37 | jaod 706 | . . 3 |
39 | 4, 38 | syl5bi 151 | . 2 |
40 | 39 | impr 376 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 wceq 1331 wcel 1480 cun 3064 wss 3066 csn 3522 (class class class)co 5767 cc 7611 cc0 7613 caddc 7616 |
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 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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-1cn 7706 ax-icn 7708 ax-addcl 7709 ax-mulcl 7711 ax-addcom 7713 ax-i2m1 7718 ax-0id 7721 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-iota 5083 df-fv 5126 df-ov 5770 |
This theorem is referenced by: nn0addcl 9005 |
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