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Theorem unss1 3151
 Description: Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
unss1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Proof of Theorem unss1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssel 3002 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21orim1d 734 . . 3 (𝐴𝐵 → ((𝑥𝐴𝑥𝐶) → (𝑥𝐵𝑥𝐶)))
3 elun 3123 . . 3 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
4 elun 3123 . . 3 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
52, 3, 43imtr4g 203 . 2 (𝐴𝐵 → (𝑥 ∈ (𝐴𝐶) → 𝑥 ∈ (𝐵𝐶)))
65ssrdv 3014 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 662   ∈ wcel 1434   ∪ cun 2980   ⊆ wss 2982 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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995 This theorem is referenced by:  unss2  3153  unss12  3154  undif1ss  3334  eldifpw  4254  tposss  5915  dftpos4  5932
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