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Theorem ssun1 3161
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 668 . . 3 (𝑥𝐴 → (𝑥𝐴𝑥𝐵))
2 elun 3139 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
31, 2sylibr 132 . 2 (𝑥𝐴𝑥 ∈ (𝐴𝐵))
43ssriv 3027 1 𝐴 ⊆ (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wo 664  wcel 1438  cun 2995  wss 2997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010
This theorem is referenced by:  ssun2  3162  ssun3  3163  elun1  3165  inabs  3229  reuun1  3279  un00  3326  undifabs  3356  undifss  3359  snsspr1  3580  snsstp1  3582  snsstp2  3583  prsstp12  3585  exmidundif  4026  sssucid  4233  unexb  4258  dmexg  4685  fvun1  5354  dftpos2  6008  tpostpos2  6012  ac6sfi  6594  caserel  6757  finomni  6775  ressxr  7510  nnssnn0  8646  un0addcl  8676  un0mulcl  8677  nn0ssxnn0  8709  fsumsplit  10764  fsum2d  10792  fsumabs  10822  bdunexb  11468
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