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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 7025 | . . 3 ⊔ inl inr | |
2 | simpr 109 | . . . . . . 7 inl inl | |
3 | df-inl 7003 | . . . . . . . . 9 inl | |
4 | opeq2 3753 | . . . . . . . . 9 | |
5 | elex 2732 | . . . . . . . . 9 | |
6 | 0ex 4103 | . . . . . . . . . . 11 | |
7 | vex 2724 | . . . . . . . . . . 11 | |
8 | 6, 7 | opex 4201 | . . . . . . . . . 10 |
9 | 8 | a1i 9 | . . . . . . . . 9 |
10 | 3, 4, 5, 9 | fvmptd3 5573 | . . . . . . . 8 inl |
11 | 10 | adantr 274 | . . . . . . 7 inl inl |
12 | 2, 11 | eqtrd 2197 | . . . . . 6 inl |
13 | elun1 3284 | . . . . . . . . 9 | |
14 | 6 | prid1 3676 | . . . . . . . . 9 |
15 | 13, 14 | jctil 310 | . . . . . . . 8 |
16 | 15 | adantr 274 | . . . . . . 7 inl |
17 | opelxp 4628 | . . . . . . 7 | |
18 | 16, 17 | sylibr 133 | . . . . . 6 inl |
19 | 12, 18 | eqeltrd 2241 | . . . . 5 inl |
20 | 19 | rexlimiva 2576 | . . . 4 inl |
21 | simpr 109 | . . . . . . 7 inr inr | |
22 | df-inr 7004 | . . . . . . . . 9 inr | |
23 | opeq2 3753 | . . . . . . . . 9 | |
24 | elex 2732 | . . . . . . . . 9 | |
25 | 1oex 6383 | . . . . . . . . . . 11 | |
26 | 25, 7 | opex 4201 | . . . . . . . . . 10 |
27 | 26 | a1i 9 | . . . . . . . . 9 |
28 | 22, 23, 24, 27 | fvmptd3 5573 | . . . . . . . 8 inr |
29 | 28 | adantr 274 | . . . . . . 7 inr inr |
30 | 21, 29 | eqtrd 2197 | . . . . . 6 inr |
31 | elun2 3285 | . . . . . . . . 9 | |
32 | 31 | adantr 274 | . . . . . . . 8 inr |
33 | 25 | prid2 3677 | . . . . . . . 8 |
34 | 32, 33 | jctil 310 | . . . . . . 7 inr |
35 | opelxp 4628 | . . . . . . 7 | |
36 | 34, 35 | sylibr 133 | . . . . . 6 inr |
37 | 30, 36 | eqeltrd 2241 | . . . . 5 inr |
38 | 37 | rexlimiva 2576 | . . . 4 inr |
39 | 20, 38 | jaoi 706 | . . 3 inl inr |
40 | 1, 39 | sylbi 120 | . 2 ⊔ |
41 | 40 | ssriv 3141 | 1 ⊔ |
Colors of variables: wff set class |
Syntax hints: wa 103 wo 698 wceq 1342 wcel 2135 wrex 2443 cvv 2721 cun 3109 wss 3111 c0 3404 cpr 3571 cop 3573 cxp 4596 cfv 5182 c1o 6368 ⊔ cdju 6993 inlcinl 7001 inrcinr 7002 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-1st 6100 df-2nd 6101 df-1o 6375 df-dju 6994 df-inl 7003 df-inr 7004 |
This theorem is referenced by: eldju1st 7027 |
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