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Theorem uniss 3648
Description: Subclass relationship for class union. Theorem 61 of [Suppes] p. 39. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
uniss (𝐴𝐵 𝐴 𝐵)

Proof of Theorem uniss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3004 . . . . 5 (𝐴𝐵 → (𝑦𝐴𝑦𝐵))
21anim2d 330 . . . 4 (𝐴𝐵 → ((𝑥𝑦𝑦𝐴) → (𝑥𝑦𝑦𝐵)))
32eximdv 1803 . . 3 (𝐴𝐵 → (∃𝑦(𝑥𝑦𝑦𝐴) → ∃𝑦(𝑥𝑦𝑦𝐵)))
4 eluni 3630 . . 3 (𝑥 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
5 eluni 3630 . . 3 (𝑥 𝐵 ↔ ∃𝑦(𝑥𝑦𝑦𝐵))
63, 4, 53imtr4g 203 . 2 (𝐴𝐵 → (𝑥 𝐴𝑥 𝐵))
76ssrdv 3016 1 (𝐴𝐵 𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wex 1422  wcel 1434  wss 2984   cuni 3627
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 2614  df-in 2990  df-ss 2997  df-uni 3628
This theorem is referenced by:  unissi  3650  unissd  3651  intssuni2m  3686  relfld  4913
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