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Theorem uniss2 4862
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 4964 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 4852 . . . . 5 ((𝑥𝑦𝑦𝐵) → 𝑥 𝐵)
21expcom 414 . . . 4 (𝑦𝐵 → (𝑥𝑦𝑥 𝐵))
32rexlimiv 3277 . . 3 (∃𝑦𝐵 𝑥𝑦𝑥 𝐵)
43ralimi 3157 . 2 (∀𝑥𝐴𝑦𝐵 𝑥𝑦 → ∀𝑥𝐴 𝑥 𝐵)
5 unissb 4861 . 2 ( 𝐴 𝐵 ↔ ∀𝑥𝐴 𝑥 𝐵)
64, 5sylibr 235 1 (∀𝑥𝐴𝑦𝐵 𝑥𝑦 𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  wral 3135  wrex 3136  wss 3933   cuni 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-v 3494  df-in 3940  df-ss 3949  df-uni 4831
This theorem is referenced by:  unidif  4863  coflim  9671
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