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Mirrors > Home > ILE Home > Th. List > elun2 | GIF version |
Description: Membership law for union of classes. (Contributed by NM, 30-Aug-1993.) |
Ref | Expression |
---|---|
elun2 | ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (𝐶 ∪ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun2 3314 | . 2 ⊢ 𝐵 ⊆ (𝐶 ∪ 𝐵) | |
2 | 1 | sseli 3166 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (𝐶 ∪ 𝐵)) |
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 dftpos4 6289 tfrlemibxssdm 6353 tfrlemi14d 6359 tfr1onlembxssdm 6369 tfr1onlemres 6375 tfrcllembxssdm 6382 tfrcllemres 6388 dcdifsnid 6530 findcard2d 6920 findcard2sd 6921 onunsnss 6946 undifdcss 6952 fisseneq 6961 fidcenumlemrks 6983 djurclr 7080 djurcl 7082 djuss 7100 finomni 7169 mnfxr 8045 hashinfuni 10792 fsumsplitsnun 11462 sumsplitdc 11475 modfsummodlem1 11499 exmidunben 12480 srnginvld 12664 lmodvscad 12682 ipsscad 12694 ipsvscad 12695 ipsipd 12696 |
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