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Mirrors > Home > ILE Home > Th. List > ssun1 | GIF version |
Description: Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
ssun1 | ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orc 712 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) | |
2 | elun 3278 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) | |
3 | 1, 2 | sylibr 134 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 ∪ 𝐵)) |
4 | 3 | ssriv 3161 | 1 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∨ wo 708 ∈ wcel 2148 ∪ cun 3129 ⊆ wss 3131 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2741 df-un 3135 df-in 3137 df-ss 3144 |
This theorem is referenced by: ssun2 3301 ssun3 3302 elun1 3304 inabs 3369 reuun1 3419 un00 3471 undifabs 3501 undifss 3505 snsspr1 3742 snsstp1 3744 snsstp2 3745 prsstp12 3747 exmidundif 4208 sssucid 4417 unexb 4444 dmexg 4893 fvun1 5584 dftpos2 6264 tpostpos2 6268 ac6sfi 6900 caserel 7088 finomni 7140 ressxr 8003 nnssnn0 9181 un0addcl 9211 un0mulcl 9212 nn0ssxnn0 9244 fsumsplit 11417 fsum2d 11445 fsumabs 11475 fprodsplitdc 11606 fprod2d 11633 ennnfonelemss 12413 lspun 13493 cnfldbas 13544 cnfldadd 13545 cnfldmul 13546 lgsdir2lem3 14516 bdunexb 14757 |
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