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Mirrors > Home > ILE Home > Th. List > elun1 | GIF version |
Description: Membership law for union of classes. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
elun1 | ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (𝐵 ∪ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 3313 | . 2 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐶) | |
2 | 1 | sseli 3166 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (𝐵 ∪ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2160 ∪ cun 3142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 |
This theorem is referenced by: dcun 3548 exmidundif 4224 exmidundifim 4225 brtposg 6278 dftpos4 6287 dcdifsnid 6528 undifdcss 6950 fidcenumlemrks 6981 djulclr 7077 djulcl 7079 djuss 7098 finomni 7167 hashennnuni 10790 sumsplitdc 11471 srngbased 12655 srngplusgd 12656 srngmulrd 12657 lmodbased 12673 lmodplusgd 12674 lmodscad 12675 ipsbased 12685 ipsaddgd 12686 ipsmulrd 12687 psrbasg 13948 |
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