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Mirrors > Home > ILE Home > Th. List > djuss | Unicode version |
Description: A disjoint union is a subset of a Cartesian product. (Contributed by AV, 25-Jun-2022.) |
Ref | Expression |
---|---|
djuss | ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djur 6922 | . . 3 ⊔ inl inr | |
2 | simpr 109 | . . . . . . 7 inl inl | |
3 | df-inl 6900 | . . . . . . . . 9 inl | |
4 | opeq2 3676 | . . . . . . . . 9 | |
5 | elex 2671 | . . . . . . . . 9 | |
6 | 0ex 4025 | . . . . . . . . . . 11 | |
7 | vex 2663 | . . . . . . . . . . 11 | |
8 | 6, 7 | opex 4121 | . . . . . . . . . 10 |
9 | 8 | a1i 9 | . . . . . . . . 9 |
10 | 3, 4, 5, 9 | fvmptd3 5482 | . . . . . . . 8 inl |
11 | 10 | adantr 274 | . . . . . . 7 inl inl |
12 | 2, 11 | eqtrd 2150 | . . . . . 6 inl |
13 | elun1 3213 | . . . . . . . . 9 | |
14 | 6 | prid1 3599 | . . . . . . . . 9 |
15 | 13, 14 | jctil 310 | . . . . . . . 8 |
16 | 15 | adantr 274 | . . . . . . 7 inl |
17 | opelxp 4539 | . . . . . . 7 | |
18 | 16, 17 | sylibr 133 | . . . . . 6 inl |
19 | 12, 18 | eqeltrd 2194 | . . . . 5 inl |
20 | 19 | rexlimiva 2521 | . . . 4 inl |
21 | simpr 109 | . . . . . . 7 inr inr | |
22 | df-inr 6901 | . . . . . . . . 9 inr | |
23 | opeq2 3676 | . . . . . . . . 9 | |
24 | elex 2671 | . . . . . . . . 9 | |
25 | 1oex 6289 | . . . . . . . . . . 11 | |
26 | 25, 7 | opex 4121 | . . . . . . . . . 10 |
27 | 26 | a1i 9 | . . . . . . . . 9 |
28 | 22, 23, 24, 27 | fvmptd3 5482 | . . . . . . . 8 inr |
29 | 28 | adantr 274 | . . . . . . 7 inr inr |
30 | 21, 29 | eqtrd 2150 | . . . . . 6 inr |
31 | elun2 3214 | . . . . . . . . 9 | |
32 | 31 | adantr 274 | . . . . . . . 8 inr |
33 | 25 | prid2 3600 | . . . . . . . 8 |
34 | 32, 33 | jctil 310 | . . . . . . 7 inr |
35 | opelxp 4539 | . . . . . . 7 | |
36 | 34, 35 | sylibr 133 | . . . . . 6 inr |
37 | 30, 36 | eqeltrd 2194 | . . . . 5 inr |
38 | 37 | rexlimiva 2521 | . . . 4 inr |
39 | 20, 38 | jaoi 690 | . . 3 inl inr |
40 | 1, 39 | sylbi 120 | . 2 ⊔ |
41 | 40 | ssriv 3071 | 1 ⊔ |
Colors of variables: wff set class |
Syntax hints: wa 103 wo 682 wceq 1316 wcel 1465 wrex 2394 cvv 2660 cun 3039 wss 3041 c0 3333 cpr 3498 cop 3500 cxp 4507 cfv 5093 c1o 6274 ⊔ cdju 6890 inlcinl 6898 inrcinr 6899 |
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-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 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-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-csb 2976 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-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-1st 6006 df-2nd 6007 df-1o 6281 df-dju 6891 df-inl 6900 df-inr 6901 |
This theorem is referenced by: eldju1st 6924 |
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