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Theorem ssun1 3209
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 686 . . 3 (𝑥𝐴 → (𝑥𝐴𝑥𝐵))
2 elun 3187 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
31, 2sylibr 133 . 2 (𝑥𝐴𝑥 ∈ (𝐴𝐵))
43ssriv 3071 1 𝐴 ⊆ (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wo 682  wcel 1465  cun 3039  wss 3041
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054
This theorem is referenced by:  ssun2  3210  ssun3  3211  elun1  3213  inabs  3278  reuun1  3328  un00  3379  undifabs  3409  undifss  3413  snsspr1  3638  snsstp1  3640  snsstp2  3641  prsstp12  3643  exmidundif  4099  sssucid  4307  unexb  4333  dmexg  4773  fvun1  5455  dftpos2  6126  tpostpos2  6130  ac6sfi  6760  caserel  6940  finomni  6980  ressxr  7777  nnssnn0  8948  un0addcl  8978  un0mulcl  8979  nn0ssxnn0  9011  fsumsplit  11144  fsum2d  11172  fsumabs  11202  ennnfonelemss  11850  bdunexb  13045
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