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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 4890 | . . . . . 6 | |
2 | 1 | uneq1i 3257 | . . . . 5 |
3 | 2 | uneq1i 3257 | . . . 4 |
4 | un23 3266 | . . . 4 | |
5 | 3, 4 | eqtri 2178 | . . 3 |
6 | setscom.1 | . . . . . . . 8 | |
7 | setsvala 12192 | . . . . . . . 8 sSet | |
8 | 6, 7 | mp3an2 1307 | . . . . . . 7 sSet |
9 | 8 | ad2ant2r 501 | . . . . . 6 sSet |
10 | 9 | reseq1d 4864 | . . . . 5 sSet |
11 | resundir 4879 | . . . . . 6 | |
12 | elex 2723 | . . . . . . . . . . 11 | |
13 | 12 | ad2antrl 482 | . . . . . . . . . 10 |
14 | opelxpi 4617 | . . . . . . . . . 10 | |
15 | 6, 13, 14 | sylancr 411 | . . . . . . . . 9 |
16 | opexg 4188 | . . . . . . . . . . 11 | |
17 | 6, 13, 16 | sylancr 411 | . . . . . . . . . 10 |
18 | relsng 4688 | . . . . . . . . . 10 | |
19 | 17, 18 | syl 14 | . . . . . . . . 9 |
20 | 15, 19 | mpbird 166 | . . . . . . . 8 |
21 | dmsnopg 5056 | . . . . . . . . . 10 | |
22 | 13, 21 | syl 14 | . . . . . . . . 9 |
23 | disjsn2 3622 | . . . . . . . . . . 11 | |
24 | 23 | ad2antlr 481 | . . . . . . . . . 10 |
25 | disj2 3449 | . . . . . . . . . 10 | |
26 | 24, 25 | sylib 121 | . . . . . . . . 9 |
27 | 22, 26 | eqsstrd 3164 | . . . . . . . 8 |
28 | relssres 4903 | . . . . . . . 8 | |
29 | 20, 27, 28 | syl2anc 409 | . . . . . . 7 |
30 | 29 | uneq2d 3261 | . . . . . 6 |
31 | 11, 30 | syl5eq 2202 | . . . . 5 |
32 | 10, 31 | eqtrd 2190 | . . . 4 sSet |
33 | 32 | uneq1d 3260 | . . 3 sSet |
34 | setscom.2 | . . . . . . . 8 | |
35 | setsvala 12192 | . . . . . . . 8 sSet | |
36 | 34, 35 | mp3an2 1307 | . . . . . . 7 sSet |
37 | 36 | reseq1d 4864 | . . . . . 6 sSet |
38 | 37 | ad2ant2rl 503 | . . . . 5 sSet |
39 | resundir 4879 | . . . . . 6 | |
40 | elex 2723 | . . . . . . . . . . 11 | |
41 | 40 | ad2antll 483 | . . . . . . . . . 10 |
42 | opelxpi 4617 | . . . . . . . . . 10 | |
43 | 34, 41, 42 | sylancr 411 | . . . . . . . . 9 |
44 | opexg 4188 | . . . . . . . . . . 11 | |
45 | 34, 41, 44 | sylancr 411 | . . . . . . . . . 10 |
46 | relsng 4688 | . . . . . . . . . 10 | |
47 | 45, 46 | syl 14 | . . . . . . . . 9 |
48 | 43, 47 | mpbird 166 | . . . . . . . 8 |
49 | dmsnopg 5056 | . . . . . . . . . 10 | |
50 | 41, 49 | syl 14 | . . . . . . . . 9 |
51 | ssv 3150 | . . . . . . . . . . 11 | |
52 | ssv 3150 | . . . . . . . . . . 11 | |
53 | ssconb 3240 | . . . . . . . . . . 11 | |
54 | 51, 52, 53 | mp2an 423 | . . . . . . . . . 10 |
55 | 26, 54 | sylib 121 | . . . . . . . . 9 |
56 | 50, 55 | eqsstrd 3164 | . . . . . . . 8 |
57 | relssres 4903 | . . . . . . . 8 | |
58 | 48, 56, 57 | syl2anc 409 | . . . . . . 7 |
59 | 58 | uneq2d 3261 | . . . . . 6 |
60 | 39, 59 | syl5eq 2202 | . . . . 5 |
61 | 38, 60 | eqtrd 2190 | . . . 4 sSet |
62 | 61 | uneq1d 3260 | . . 3 sSet |
63 | 5, 33, 62 | 3eqtr4a 2216 | . 2 sSet sSet |
64 | setsex 12193 | . . . . 5 sSet | |
65 | 6, 64 | mp3an2 1307 | . . . 4 sSet |
66 | 65 | ad2ant2r 501 | . . 3 sSet |
67 | 34 | a1i 9 | . . 3 |
68 | simprr 522 | . . 3 | |
69 | setsvala 12192 | . . 3 sSet sSet sSet sSet | |
70 | 66, 67, 68, 69 | syl3anc 1220 | . 2 sSet sSet sSet |
71 | setsex 12193 | . . . . 5 sSet | |
72 | 34, 71 | mp3an2 1307 | . . . 4 sSet |
73 | 72 | ad2ant2rl 503 | . . 3 sSet |
74 | 6 | a1i 9 | . . 3 |
75 | simprl 521 | . . 3 | |
76 | setsvala 12192 | . . 3 sSet sSet sSet sSet | |
77 | 73, 74, 75, 76 | syl3anc 1220 | . 2 sSet sSet sSet |
78 | 63, 70, 77 | 3eqtr4d 2200 | 1 sSet sSet sSet sSet |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wcel 2128 wne 2327 cvv 2712 cdif 3099 cun 3100 cin 3101 wss 3102 c0 3394 csn 3560 cop 3563 cxp 4583 cdm 4585 cres 4587 wrel 4590 (class class class)co 5821 sSet csts 12159 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-id 4253 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-res 4597 df-iota 5134 df-fun 5171 df-fv 5177 df-ov 5824 df-oprab 5825 df-mpo 5826 df-sets 12168 |
This theorem is referenced by: (None) |
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