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Mirrors > Home > ILE Home > Th. List > djulcl | Unicode version |
Description: Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) |
Ref | Expression |
---|---|
djulcl | inl ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2692 | . . 3 | |
2 | 0ex 4050 | . . . . 5 | |
3 | 2 | snid 3551 | . . . 4 |
4 | opelxpi 4566 | . . . 4 | |
5 | 3, 4 | mpan 420 | . . 3 |
6 | opeq2 3701 | . . . 4 | |
7 | df-inl 6925 | . . . 4 inl | |
8 | 6, 7 | fvmptg 5490 | . . 3 inl |
9 | 1, 5, 8 | syl2anc 408 | . 2 inl |
10 | elun1 3238 | . . . 4 | |
11 | 5, 10 | syl 14 | . . 3 |
12 | df-dju 6916 | . . 3 ⊔ | |
13 | 11, 12 | eleqtrrdi 2231 | . 2 ⊔ |
14 | 9, 13 | eqeltrd 2214 | 1 inl ⊔ |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 wcel 1480 cvv 2681 cun 3064 c0 3358 csn 3522 cop 3525 cxp 4532 cfv 5118 c1o 6299 ⊔ cdju 6915 inlcinl 6923 |
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 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-dju 6916 df-inl 6925 |
This theorem is referenced by: djulclb 6933 updjudhcoinlf 6958 omp1eomlem 6972 difinfsnlem 6977 difinfsn 6978 ctmlemr 6986 ctm 6987 ctssdclemn0 6988 ctssdccl 6989 fodju0 7012 exmidfodomrlemr 7051 exmidfodomrlemrALT 7052 subctctexmid 13185 |
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