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Mirrors > Home > ILE Home > Th. List > setscom | Unicode version |
Description: Component-setting is commutative when the x-values are different. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
setscom.1 | |
setscom.2 |
Ref | Expression |
---|---|
setscom | sSet sSet sSet sSet |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rescom 4814 | . . . . . 6 | |
2 | 1 | uneq1i 3196 | . . . . 5 |
3 | 2 | uneq1i 3196 | . . . 4 |
4 | un23 3205 | . . . 4 | |
5 | 3, 4 | eqtri 2138 | . . 3 |
6 | setscom.1 | . . . . . . . 8 | |
7 | setsvala 11917 | . . . . . . . 8 sSet | |
8 | 6, 7 | mp3an2 1288 | . . . . . . 7 sSet |
9 | 8 | ad2ant2r 500 | . . . . . 6 sSet |
10 | 9 | reseq1d 4788 | . . . . 5 sSet |
11 | resundir 4803 | . . . . . 6 | |
12 | elex 2671 | . . . . . . . . . . 11 | |
13 | 12 | ad2antrl 481 | . . . . . . . . . 10 |
14 | opelxpi 4541 | . . . . . . . . . 10 | |
15 | 6, 13, 14 | sylancr 410 | . . . . . . . . 9 |
16 | opexg 4120 | . . . . . . . . . . 11 | |
17 | 6, 13, 16 | sylancr 410 | . . . . . . . . . 10 |
18 | relsng 4612 | . . . . . . . . . 10 | |
19 | 17, 18 | syl 14 | . . . . . . . . 9 |
20 | 15, 19 | mpbird 166 | . . . . . . . 8 |
21 | dmsnopg 4980 | . . . . . . . . . 10 | |
22 | 13, 21 | syl 14 | . . . . . . . . 9 |
23 | disjsn2 3556 | . . . . . . . . . . 11 | |
24 | 23 | ad2antlr 480 | . . . . . . . . . 10 |
25 | disj2 3388 | . . . . . . . . . 10 | |
26 | 24, 25 | sylib 121 | . . . . . . . . 9 |
27 | 22, 26 | eqsstrd 3103 | . . . . . . . 8 |
28 | relssres 4827 | . . . . . . . 8 | |
29 | 20, 27, 28 | syl2anc 408 | . . . . . . 7 |
30 | 29 | uneq2d 3200 | . . . . . 6 |
31 | 11, 30 | syl5eq 2162 | . . . . 5 |
32 | 10, 31 | eqtrd 2150 | . . . 4 sSet |
33 | 32 | uneq1d 3199 | . . 3 sSet |
34 | setscom.2 | . . . . . . . 8 | |
35 | setsvala 11917 | . . . . . . . 8 sSet | |
36 | 34, 35 | mp3an2 1288 | . . . . . . 7 sSet |
37 | 36 | reseq1d 4788 | . . . . . 6 sSet |
38 | 37 | ad2ant2rl 502 | . . . . 5 sSet |
39 | resundir 4803 | . . . . . 6 | |
40 | elex 2671 | . . . . . . . . . . 11 | |
41 | 40 | ad2antll 482 | . . . . . . . . . 10 |
42 | opelxpi 4541 | . . . . . . . . . 10 | |
43 | 34, 41, 42 | sylancr 410 | . . . . . . . . 9 |
44 | opexg 4120 | . . . . . . . . . . 11 | |
45 | 34, 41, 44 | sylancr 410 | . . . . . . . . . 10 |
46 | relsng 4612 | . . . . . . . . . 10 | |
47 | 45, 46 | syl 14 | . . . . . . . . 9 |
48 | 43, 47 | mpbird 166 | . . . . . . . 8 |
49 | dmsnopg 4980 | . . . . . . . . . 10 | |
50 | 41, 49 | syl 14 | . . . . . . . . 9 |
51 | ssv 3089 | . . . . . . . . . . 11 | |
52 | ssv 3089 | . . . . . . . . . . 11 | |
53 | ssconb 3179 | . . . . . . . . . . 11 | |
54 | 51, 52, 53 | mp2an 422 | . . . . . . . . . 10 |
55 | 26, 54 | sylib 121 | . . . . . . . . 9 |
56 | 50, 55 | eqsstrd 3103 | . . . . . . . 8 |
57 | relssres 4827 | . . . . . . . 8 | |
58 | 48, 56, 57 | syl2anc 408 | . . . . . . 7 |
59 | 58 | uneq2d 3200 | . . . . . 6 |
60 | 39, 59 | syl5eq 2162 | . . . . 5 |
61 | 38, 60 | eqtrd 2150 | . . . 4 sSet |
62 | 61 | uneq1d 3199 | . . 3 sSet |
63 | 5, 33, 62 | 3eqtr4a 2176 | . 2 sSet sSet |
64 | setsex 11918 | . . . . 5 sSet | |
65 | 6, 64 | mp3an2 1288 | . . . 4 sSet |
66 | 65 | ad2ant2r 500 | . . 3 sSet |
67 | 34 | a1i 9 | . . 3 |
68 | simprr 506 | . . 3 | |
69 | setsvala 11917 | . . 3 sSet sSet sSet sSet | |
70 | 66, 67, 68, 69 | syl3anc 1201 | . 2 sSet sSet sSet |
71 | setsex 11918 | . . . . 5 sSet | |
72 | 34, 71 | mp3an2 1288 | . . . 4 sSet |
73 | 72 | ad2ant2rl 502 | . . 3 sSet |
74 | 6 | a1i 9 | . . 3 |
75 | simprl 505 | . . 3 | |
76 | setsvala 11917 | . . 3 sSet sSet sSet sSet | |
77 | 73, 74, 75, 76 | syl3anc 1201 | . 2 sSet sSet sSet |
78 | 63, 70, 77 | 3eqtr4d 2160 | 1 sSet sSet sSet sSet |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 wne 2285 cvv 2660 cdif 3038 cun 3039 cin 3040 wss 3041 c0 3333 csn 3497 cop 3500 cxp 4507 cdm 4509 cres 4511 wrel 4514 (class class class)co 5742 sSet csts 11884 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-res 4521 df-iota 5058 df-fun 5095 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-sets 11893 |
This theorem is referenced by: (None) |
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