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Theorem uniss2 4833
 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 4936 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 4825 . . . . 5 ((𝑥𝑦𝑦𝐵) → 𝑥 𝐵)
21expcom 417 . . . 4 (𝑦𝐵 → (𝑥𝑦𝑥 𝐵))
32rexlimiv 3239 . . 3 (∃𝑦𝐵 𝑥𝑦𝑥 𝐵)
43ralimi 3128 . 2 (∀𝑥𝐴𝑦𝐵 𝑥𝑦 → ∀𝑥𝐴 𝑥 𝐵)
5 unissb 4832 . 2 ( 𝐴 𝐵 ↔ ∀𝑥𝐴 𝑥 𝐵)
64, 5sylibr 237 1 (∀𝑥𝐴𝑦𝐵 𝑥𝑦 𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ⊆ wss 3881  ∪ cuni 4800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112  df-v 3443  df-in 3888  df-ss 3898  df-uni 4801 This theorem is referenced by:  unidif  4834  coflim  9672
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