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Theorem ssun1 3134
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 643 . . 3 (𝑥𝐴 → (𝑥𝐴𝑥𝐵))
2 elun 3112 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
31, 2sylibr 141 . 2 (𝑥𝐴𝑥 ∈ (𝐴𝐵))
43ssriv 2977 1 𝐴 ⊆ (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wo 639  wcel 1409  cun 2943  wss 2945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959
This theorem is referenced by:  ssun2  3135  ssun3  3136  elun1  3138  inabs  3196  reuun1  3247  un00  3291  undifabs  3328  undifss  3331  snsspr1  3540  snsstp1  3542  snsstp2  3543  prsstp12  3545  sssucid  4180  unexb  4205  dmexg  4624  fvun1  5267  dftpos2  5907  tpostpos2  5911  ac6sfi  6383  ressxr  7128  nnssnn0  8242  un0addcl  8272  un0mulcl  8273  bdunexb  10427
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