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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 7020 | . 2 ⊔ | |
2 | df-dju 7015 | . . . . 5 ⊔ | |
3 | 2 | eleq1i 2236 | . . . 4 ⊔ |
4 | unexb 4427 | . . . 4 | |
5 | 3, 4 | bitr4i 186 | . . 3 ⊔ |
6 | 0ex 4116 | . . . . . . 7 | |
7 | 6 | snm 3703 | . . . . . 6 |
8 | rnxpm 5040 | . . . . . 6 | |
9 | 7, 8 | ax-mp 5 | . . . . 5 |
10 | rnexg 4876 | . . . . 5 | |
11 | 9, 10 | eqeltrrid 2258 | . . . 4 |
12 | 1oex 6403 | . . . . . . 7 | |
13 | 12 | snm 3703 | . . . . . 6 |
14 | rnxpm 5040 | . . . . . 6 | |
15 | 13, 14 | ax-mp 5 | . . . . 5 |
16 | rnexg 4876 | . . . . 5 | |
17 | 15, 16 | eqeltrrid 2258 | . . . 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 1348 wex 1485 wcel 2141 cvv 2730 cun 3119 c0 3414 csn 3583 cxp 4609 crn 4612 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-eu 2022 df-mo 2023 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-br 3990 df-opab 4051 df-tr 4088 df-iord 4351 df-on 4353 df-suc 4356 df-xp 4617 df-rel 4618 df-cnv 4619 df-dm 4621 df-rn 4622 df-1o 6395 df-dju 7015 |
This theorem is referenced by: ctfoex 7095 |
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