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Mirrors > Home > ILE Home > Th. List > djulclb | Unicode version |
Description: Left biconditional closure of disjoint union. (Contributed by Jim Kingdon, 2-Jul-2022.) |
Ref | Expression |
---|---|
djulclb | inl ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djulcl 6936 | . 2 inl ⊔ | |
2 | 1n0 6329 | . . . . . . . . . 10 | |
3 | 2 | necomi 2393 | . . . . . . . . 9 |
4 | 0ex 4055 | . . . . . . . . . 10 | |
5 | 4 | elsn 3543 | . . . . . . . . 9 |
6 | 3, 5 | nemtbir 2397 | . . . . . . . 8 |
7 | 6 | intnanr 915 | . . . . . . 7 |
8 | opelxp 4569 | . . . . . . 7 | |
9 | 7, 8 | mtbir 660 | . . . . . 6 |
10 | elex 2697 | . . . . . . . . . . . 12 | |
11 | opexg 4150 | . . . . . . . . . . . . 13 | |
12 | 4, 11 | mpan 420 | . . . . . . . . . . . 12 |
13 | opeq2 3706 | . . . . . . . . . . . . 13 | |
14 | df-inl 6932 | . . . . . . . . . . . . 13 inl | |
15 | 13, 14 | fvmptg 5497 | . . . . . . . . . . . 12 inl |
16 | 10, 12, 15 | syl2anc 408 | . . . . . . . . . . 11 inl |
17 | 16 | adantr 274 | . . . . . . . . . 10 inl ⊔ inl |
18 | df-dju 6923 | . . . . . . . . . . . . 13 ⊔ | |
19 | 18 | eleq2i 2206 | . . . . . . . . . . . 12 inl ⊔ inl |
20 | 19 | biimpi 119 | . . . . . . . . . . 11 inl ⊔ inl |
21 | 20 | adantl 275 | . . . . . . . . . 10 inl ⊔ inl |
22 | 17, 21 | eqeltrrd 2217 | . . . . . . . . 9 inl ⊔ |
23 | elun 3217 | . . . . . . . . 9 | |
24 | 22, 23 | sylib 121 | . . . . . . . 8 inl ⊔ |
25 | 24 | orcomd 718 | . . . . . . 7 inl ⊔ |
26 | 25 | ord 713 | . . . . . 6 inl ⊔ |
27 | 9, 26 | mpi 15 | . . . . 5 inl ⊔ |
28 | opelxp 4569 | . . . . 5 | |
29 | 27, 28 | sylib 121 | . . . 4 inl ⊔ |
30 | 29 | simprd 113 | . . 3 inl ⊔ |
31 | 30 | ex 114 | . 2 inl ⊔ |
32 | 1, 31 | impbid2 142 | 1 inl ⊔ |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 wceq 1331 wcel 1480 cvv 2686 cun 3069 c0 3363 csn 3527 cop 3530 cxp 4537 cfv 5123 c1o 6306 ⊔ cdju 6922 inlcinl 6930 |
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 603 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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 |
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-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 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-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-1o 6313 df-dju 6923 df-inl 6932 |
This theorem is referenced by: exmidfodomrlemr 7058 |
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