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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 7015. (Contributed by AV, 28-Jun-2022.) |
Ref | Expression |
---|---|
djuex | ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dju 6983 | . 2 ⊔ | |
2 | p0ex 4150 | . . . . . . 7 | |
3 | 2 | a1i 9 | . . . . . 6 |
4 | 3 | anim1i 338 | . . . . 5 |
5 | 4 | ancoms 266 | . . . 4 |
6 | xpexg 4701 | . . . 4 | |
7 | 5, 6 | syl 14 | . . 3 |
8 | 1on 6371 | . . . . . . 7 | |
9 | 8 | elexi 2724 | . . . . . 6 |
10 | 9 | snex 4147 | . . . . 5 |
11 | 10 | a1i 9 | . . . 4 |
12 | xpexg 4701 | . . . 4 | |
13 | 11, 12 | sylan 281 | . . 3 |
14 | unexg 4404 | . . 3 | |
15 | 7, 13, 14 | syl2anc 409 | . 2 |
16 | 1, 15 | eqeltrid 2244 | 1 ⊔ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 2128 cvv 2712 cun 3100 c0 3394 csn 3560 con0 4324 cxp 4585 c1o 6357 ⊔ cdju 6982 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-opab 4027 df-tr 4064 df-iord 4327 df-on 4329 df-suc 4332 df-xp 4593 df-1o 6364 df-dju 6983 |
This theorem is referenced by: djuexb 6989 updjud 7027 djudom 7038 exmidfodomrlemr 7138 exmidfodomrlemrALT 7139 djudoml 7155 djudomr 7156 exmidsbthrlem 13635 sbthom 13639 |
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