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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 7047. (Contributed by AV, 28-Jun-2022.) |
Ref | Expression |
---|---|
djuex | ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dju 7015 | . 2 ⊔ | |
2 | p0ex 4174 | . . . . . . 7 | |
3 | 2 | a1i 9 | . . . . . 6 |
4 | 3 | anim1i 338 | . . . . 5 |
5 | 4 | ancoms 266 | . . . 4 |
6 | xpexg 4725 | . . . 4 | |
7 | 5, 6 | syl 14 | . . 3 |
8 | 1on 6402 | . . . . . . 7 | |
9 | 8 | elexi 2742 | . . . . . 6 |
10 | 9 | snex 4171 | . . . . 5 |
11 | 10 | a1i 9 | . . . 4 |
12 | xpexg 4725 | . . . 4 | |
13 | 11, 12 | sylan 281 | . . 3 |
14 | unexg 4428 | . . 3 | |
15 | 7, 13, 14 | syl2anc 409 | . 2 |
16 | 1, 15 | eqeltrid 2257 | 1 ⊔ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 2141 cvv 2730 cun 3119 c0 3414 csn 3583 con0 4348 cxp 4609 c1o 6388 ⊔ cdju 7014 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-opab 4051 df-tr 4088 df-iord 4351 df-on 4353 df-suc 4356 df-xp 4617 df-1o 6395 df-dju 7015 |
This theorem is referenced by: djuexb 7021 updjud 7059 djudom 7070 exmidfodomrlemr 7179 exmidfodomrlemrALT 7180 djudoml 7196 djudomr 7197 exmidsbthrlem 14054 sbthom 14058 |
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