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Theorem uniss2 4406
Description: A subclass condition on the members of two classes that implies a subclass relation on their unions. Proposition 8.6 of [TakeutiZaring] p. 59. See iunss2 4501 for a generalization to indexed unions. (Contributed by NM, 22-Mar-2004.)
Assertion
Ref Expression
uniss2 (∀𝑥𝐴𝑦𝐵 𝑥𝑦 𝐴 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem uniss2
StepHypRef Expression
1 ssuni 4395 . . . . 5 ((𝑥𝑦𝑦𝐵) → 𝑥 𝐵)
21expcom 450 . . . 4 (𝑦𝐵 → (𝑥𝑦𝑥 𝐵))
32rexlimiv 3009 . . 3 (∃𝑦𝐵 𝑥𝑦𝑥 𝐵)
43ralimi 2936 . 2 (∀𝑥𝐴𝑦𝐵 𝑥𝑦 → ∀𝑥𝐴 𝑥 𝐵)
5 unissb 4405 . 2 ( 𝐴 𝐵 ↔ ∀𝑥𝐴 𝑥 𝐵)
64, 5sylibr 223 1 (∀𝑥𝐴𝑦𝐵 𝑥𝑦 𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1977  wral 2896  wrex 2897  wss 3540   cuni 4372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554  df-uni 4373
This theorem is referenced by:  unidif  4407  coflim  8966
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