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Theorem uniss2 4282
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 4375 for a generalization to indexed unions. (Contributed by NM, 22-Mar-2004.)
Assertion
Ref Expression
uniss2
Distinct variable groups:   ,   , ,

Proof of Theorem uniss2
StepHypRef Expression
1 ssuni 4271 . . . . 5
21expcom 435 . . . 4
32rexlimiv 2943 . . 3
43ralimi 2850 . 2
5 unissb 4281 . 2
64, 5sylibr 212 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  e.wcel 1818  A.wral 2807  E.wrex 2808  C_wss 3475  U.cuni 4249
This theorem is referenced by:  unidif  4283  coflim  8662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-in 3482  df-ss 3489  df-uni 4250
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