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Mirrors > Home > ILE Home > Th. List > djuexb | Unicode version |
Description: The disjoint union of two classes is a set iff both classes are sets. (Contributed by Jim Kingdon, 6-Sep-2023.) |
Ref | Expression |
---|---|
djuexb | ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djuex 6987 | . 2 ⊔ | |
2 | df-dju 6982 | . . . . 5 ⊔ | |
3 | 2 | eleq1i 2223 | . . . 4 ⊔ |
4 | unexb 4402 | . . . 4 | |
5 | 3, 4 | bitr4i 186 | . . 3 ⊔ |
6 | 0ex 4091 | . . . . . . 7 | |
7 | 6 | snm 3679 | . . . . . 6 |
8 | rnxpm 5015 | . . . . . 6 | |
9 | 7, 8 | ax-mp 5 | . . . . 5 |
10 | rnexg 4851 | . . . . 5 | |
11 | 9, 10 | eqeltrrid 2245 | . . . 4 |
12 | 1oex 6371 | . . . . . . 7 | |
13 | 12 | snm 3679 | . . . . . 6 |
14 | rnxpm 5015 | . . . . . 6 | |
15 | 13, 14 | ax-mp 5 | . . . . 5 |
16 | rnexg 4851 | . . . . 5 | |
17 | 15, 16 | eqeltrrid 2245 | . . . 4 |
18 | 11, 17 | anim12i 336 | . . 3 |
19 | 5, 18 | sylbi 120 | . 2 ⊔ |
20 | 1, 19 | impbii 125 | 1 ⊔ |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1335 wex 1472 wcel 2128 cvv 2712 cun 3100 c0 3394 csn 3560 cxp 4584 crn 4587 c1o 6356 ⊔ cdju 6981 |
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 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 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 3773 df-br 3966 df-opab 4026 df-tr 4063 df-iord 4326 df-on 4328 df-suc 4331 df-xp 4592 df-rel 4593 df-cnv 4594 df-dm 4596 df-rn 4597 df-1o 6363 df-dju 6982 |
This theorem is referenced by: ctfoex 7062 |
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