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Mirrors > Home > ILE Home > Th. List > djuex | Unicode version |
Description: The disjoint union of sets is a set. See also the more precise djuss 6923. (Contributed by AV, 28-Jun-2022.) |
Ref | Expression |
---|---|
djuex | ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dju 6891 | . 2 ⊔ | |
2 | p0ex 4082 | . . . . . . 7 | |
3 | 2 | a1i 9 | . . . . . 6 |
4 | 3 | anim1i 338 | . . . . 5 |
5 | 4 | ancoms 266 | . . . 4 |
6 | xpexg 4623 | . . . 4 | |
7 | 5, 6 | syl 14 | . . 3 |
8 | 1on 6288 | . . . . . . 7 | |
9 | 8 | elexi 2672 | . . . . . 6 |
10 | 9 | snex 4079 | . . . . 5 |
11 | 10 | a1i 9 | . . . 4 |
12 | xpexg 4623 | . . . 4 | |
13 | 11, 12 | sylan 281 | . . 3 |
14 | unexg 4334 | . . 3 | |
15 | 7, 13, 14 | syl2anc 408 | . 2 |
16 | 1, 15 | eqeltrid 2204 | 1 ⊔ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 1465 cvv 2660 cun 3039 c0 3333 csn 3497 con0 4255 cxp 4507 c1o 6274 ⊔ cdju 6890 |
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-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 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-opab 3960 df-tr 3997 df-iord 4258 df-on 4260 df-suc 4263 df-xp 4515 df-1o 6281 df-dju 6891 |
This theorem is referenced by: djuexb 6897 updjud 6935 djudom 6946 exmidfodomrlemr 7026 exmidfodomrlemrALT 7027 djudoml 7043 djudomr 7044 exmidsbthrlem 13144 sbthom 13148 |
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