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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 7058 | . . 3 ⊔ inl inr | |
2 | simpr 110 | . . . . . . 7 inl inl | |
3 | df-inl 7036 | . . . . . . . . 9 inl | |
4 | opeq2 3775 | . . . . . . . . 9 | |
5 | elex 2746 | . . . . . . . . 9 | |
6 | 0ex 4125 | . . . . . . . . . . 11 | |
7 | vex 2738 | . . . . . . . . . . 11 | |
8 | 6, 7 | opex 4223 | . . . . . . . . . 10 |
9 | 8 | a1i 9 | . . . . . . . . 9 |
10 | 3, 4, 5, 9 | fvmptd3 5601 | . . . . . . . 8 inl |
11 | 10 | adantr 276 | . . . . . . 7 inl inl |
12 | 2, 11 | eqtrd 2208 | . . . . . 6 inl |
13 | elun1 3300 | . . . . . . . . 9 | |
14 | 6 | prid1 3695 | . . . . . . . . 9 |
15 | 13, 14 | jctil 312 | . . . . . . . 8 |
16 | 15 | adantr 276 | . . . . . . 7 inl |
17 | opelxp 4650 | . . . . . . 7 | |
18 | 16, 17 | sylibr 134 | . . . . . 6 inl |
19 | 12, 18 | eqeltrd 2252 | . . . . 5 inl |
20 | 19 | rexlimiva 2587 | . . . 4 inl |
21 | simpr 110 | . . . . . . 7 inr inr | |
22 | df-inr 7037 | . . . . . . . . 9 inr | |
23 | opeq2 3775 | . . . . . . . . 9 | |
24 | elex 2746 | . . . . . . . . 9 | |
25 | 1oex 6415 | . . . . . . . . . . 11 | |
26 | 25, 7 | opex 4223 | . . . . . . . . . 10 |
27 | 26 | a1i 9 | . . . . . . . . 9 |
28 | 22, 23, 24, 27 | fvmptd3 5601 | . . . . . . . 8 inr |
29 | 28 | adantr 276 | . . . . . . 7 inr inr |
30 | 21, 29 | eqtrd 2208 | . . . . . 6 inr |
31 | elun2 3301 | . . . . . . . . 9 | |
32 | 31 | adantr 276 | . . . . . . . 8 inr |
33 | 25 | prid2 3696 | . . . . . . . 8 |
34 | 32, 33 | jctil 312 | . . . . . . 7 inr |
35 | opelxp 4650 | . . . . . . 7 | |
36 | 34, 35 | sylibr 134 | . . . . . 6 inr |
37 | 30, 36 | eqeltrd 2252 | . . . . 5 inr |
38 | 37 | rexlimiva 2587 | . . . 4 inr |
39 | 20, 38 | jaoi 716 | . . 3 inl inr |
40 | 1, 39 | sylbi 121 | . 2 ⊔ |
41 | 40 | ssriv 3157 | 1 ⊔ |
Colors of variables: wff set class |
Syntax hints: wa 104 wo 708 wceq 1353 wcel 2146 wrex 2454 cvv 2735 cun 3125 wss 3127 c0 3420 cpr 3590 cop 3592 cxp 4618 cfv 5208 c1o 6400 ⊔ cdju 7026 inlcinl 7034 inrcinr 7035 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-iord 4360 df-on 4362 df-suc 4365 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-1st 6131 df-2nd 6132 df-1o 6407 df-dju 7027 df-inl 7036 df-inr 7037 |
This theorem is referenced by: eldju1st 7060 |
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