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Mirrors > Home > ILE Home > Th. List > djuen | Unicode version |
Description: Disjoint unions of equinumerous sets are equinumerous. (Contributed by Jim Kingdon, 30-Jul-2023.) |
Ref | Expression |
---|---|
djuen | ⊔ ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | encv 6706 | . . . . . . . 8 | |
2 | 1 | adantr 274 | . . . . . . 7 |
3 | 2 | simpld 111 | . . . . . 6 |
4 | eninl 7056 | . . . . . 6 inl | |
5 | 3, 4 | syl 14 | . . . . 5 inl |
6 | simpl 108 | . . . . 5 | |
7 | entr 6744 | . . . . 5 inl inl | |
8 | 5, 6, 7 | syl2anc 409 | . . . 4 inl |
9 | eninl 7056 | . . . . . 6 inl | |
10 | 2, 9 | simpl2im 384 | . . . . 5 inl |
11 | 10 | ensymd 6743 | . . . 4 inl |
12 | entr 6744 | . . . 4 inl inl inl inl | |
13 | 8, 11, 12 | syl2anc 409 | . . 3 inl inl |
14 | encv 6706 | . . . . . . . 8 | |
15 | 14 | adantl 275 | . . . . . . 7 |
16 | 15 | simpld 111 | . . . . . 6 |
17 | eninr 7057 | . . . . . 6 inr | |
18 | 16, 17 | syl 14 | . . . . 5 inr |
19 | entr 6744 | . . . . 5 inr inr | |
20 | 18, 19 | sylancom 417 | . . . 4 inr |
21 | eninr 7057 | . . . . . 6 inr | |
22 | 15, 21 | simpl2im 384 | . . . . 5 inr |
23 | 22 | ensymd 6743 | . . . 4 inr |
24 | entr 6744 | . . . 4 inr inr inr inr | |
25 | 20, 23, 24 | syl2anc 409 | . . 3 inr inr |
26 | djuin 7023 | . . . 4 inl inr | |
27 | 26 | a1i 9 | . . 3 inl inr |
28 | djuin 7023 | . . . 4 inl inr | |
29 | 28 | a1i 9 | . . 3 inl inr |
30 | unen 6776 | . . 3 inl inl inr inr inl inr inl inr inl inr inl inr | |
31 | 13, 25, 27, 29, 30 | syl22anc 1228 | . 2 inl inr inl inr |
32 | djuun 7026 | . 2 inl inr ⊔ | |
33 | djuun 7026 | . 2 inl inr ⊔ | |
34 | 31, 32, 33 | 3brtr3g 4012 | 1 ⊔ ⊔ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 cvv 2724 cun 3112 cin 3113 c0 3407 class class class wbr 3979 cima 4604 cen 6698 ⊔ cdju 6996 inlcinl 7004 inrcinr 7005 |
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-coll 4094 ax-sep 4097 ax-nul 4105 ax-pow 4150 ax-pr 4184 ax-un 4408 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 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-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2726 df-sbc 2950 df-csb 3044 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-nul 3408 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-iun 3865 df-br 3980 df-opab 4041 df-mpt 4042 df-tr 4078 df-id 4268 df-iord 4341 df-on 4343 df-suc 4346 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-rn 4612 df-res 4613 df-ima 4614 df-iota 5150 df-fun 5187 df-fn 5188 df-f 5189 df-f1 5190 df-fo 5191 df-f1o 5192 df-fv 5193 df-1st 6103 df-2nd 6104 df-1o 6378 df-er 6495 df-en 6701 df-dju 6997 df-inl 7006 df-inr 7007 |
This theorem is referenced by: djuenun 7162 exmidunben 12353 enctlem 12359 |
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