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Theorem ssun1 3300
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 712 . . 3 (𝑥𝐴 → (𝑥𝐴𝑥𝐵))
2 elun 3278 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
31, 2sylibr 134 . 2 (𝑥𝐴𝑥 ∈ (𝐴𝐵))
43ssriv 3161 1 𝐴 ⊆ (𝐴𝐵)
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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