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Mirrors > Home > ILE Home > Th. List > eldju2ndr | Unicode version |
Description: The second component of an element of a disjoint union is an element of the right class of the disjoint union if its first component is not the empty set. (Contributed by AV, 26-Jun-2022.) |
Ref | Expression |
---|---|
eldju2ndr | ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dju 6923 | . . . . 5 ⊔ | |
2 | 1 | eleq2i 2206 | . . . 4 ⊔ |
3 | elun 3217 | . . . 4 | |
4 | 2, 3 | bitri 183 | . . 3 ⊔ |
5 | elxp6 6067 | . . . . 5 | |
6 | elsni 3545 | . . . . . . 7 | |
7 | eqneqall 2318 | . . . . . . 7 | |
8 | 6, 7 | syl 14 | . . . . . 6 |
9 | 8 | ad2antrl 481 | . . . . 5 |
10 | 5, 9 | sylbi 120 | . . . 4 |
11 | elxp6 6067 | . . . . 5 | |
12 | simprr 521 | . . . . . 6 | |
13 | 12 | a1d 22 | . . . . 5 |
14 | 11, 13 | sylbi 120 | . . . 4 |
15 | 10, 14 | jaoi 705 | . . 3 |
16 | 4, 15 | sylbi 120 | . 2 ⊔ |
17 | 16 | imp 123 | 1 ⊔ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 wceq 1331 wcel 1480 wne 2308 cun 3069 c0 3363 csn 3527 cop 3530 cxp 4537 cfv 5123 c1st 6036 c2nd 6037 c1o 6306 ⊔ cdju 6922 |
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-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fv 5131 df-1st 6038 df-2nd 6039 df-dju 6923 |
This theorem is referenced by: updjudhf 6964 |
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