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Theorem uniss 3669
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 3017 . . . . 5 (𝐴𝐵 → (𝑦𝐴𝑦𝐵))
21anim2d 330 . . . 4 (𝐴𝐵 → ((𝑥𝑦𝑦𝐴) → (𝑥𝑦𝑦𝐵)))
32eximdv 1808 . . 3 (𝐴𝐵 → (∃𝑦(𝑥𝑦𝑦𝐴) → ∃𝑦(𝑥𝑦𝑦𝐵)))
4 eluni 3651 . . 3 (𝑥 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
5 eluni 3651 . . 3 (𝑥 𝐵 ↔ ∃𝑦(𝑥𝑦𝑦𝐵))
63, 4, 53imtr4g 203 . 2 (𝐴𝐵 → (𝑥 𝐴𝑥 𝐵))
76ssrdv 3029 1 (𝐴𝐵 𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wex 1426  wcel 1438  wss 2997   cuni 3648
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003  df-ss 3010  df-uni 3649
This theorem is referenced by:  unissi  3671  unissd  3672  intssuni2m  3707  relfld  4946
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