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Theorem uniss2 4944
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 5051 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 4935 . . . . 5 ((𝑥𝑦𝑦𝐵) → 𝑥 𝐵)
21expcom 415 . . . 4 (𝑦𝐵 → (𝑥𝑦𝑥 𝐵))
32rexlimiv 3149 . . 3 (∃𝑦𝐵 𝑥𝑦𝑥 𝐵)
43ralimi 3084 . 2 (∀𝑥𝐴𝑦𝐵 𝑥𝑦 → ∀𝑥𝐴 𝑥 𝐵)
5 unissb 4942 . 2 ( 𝐴 𝐵 ↔ ∀𝑥𝐴 𝑥 𝐵)
64, 5sylibr 233 1 (∀𝑥𝐴𝑦𝐵 𝑥𝑦 𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wral 3062  wrex 3071  wss 3947   cuni 4907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-v 3477  df-in 3954  df-ss 3964  df-uni 4908
This theorem is referenced by:  unidif  4945  coflim  10252  onsssupeqcond  41963
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