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Theorem ssun1 3244
 Description: Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
ssun1 𝐴 ⊆ (𝐴𝐵)

Proof of Theorem ssun1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 orc 702 . . 3 (𝑥𝐴 → (𝑥𝐴𝑥𝐵))
2 elun 3222 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
31, 2sylibr 133 . 2 (𝑥𝐴𝑥 ∈ (𝐴𝐵))
43ssriv 3106 1 𝐴 ⊆ (𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   ∨ wo 698   ∈ wcel 1481   ∪ cun 3074   ⊆ wss 3076 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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089 This theorem is referenced by:  ssun2  3245  ssun3  3246  elun1  3248  inabs  3313  reuun1  3363  un00  3414  undifabs  3444  undifss  3448  snsspr1  3676  snsstp1  3678  snsstp2  3679  prsstp12  3681  exmidundif  4137  sssucid  4345  unexb  4371  dmexg  4811  fvun1  5495  dftpos2  6166  tpostpos2  6170  ac6sfi  6800  caserel  6980  finomni  7020  ressxr  7833  nnssnn0  9004  un0addcl  9034  un0mulcl  9035  nn0ssxnn0  9067  fsumsplit  11208  fsum2d  11236  fsumabs  11266  ennnfonelemss  11959  bdunexb  13289
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