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Theorem uniss2 3602
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. (Contributed by NM, 22-Mar-2004.)
Assertion
Ref Expression
uniss2 (x A y B xy A B)
Distinct variable groups:   x,A   x,y,B
Allowed substitution hint:   A(y)

Proof of Theorem uniss2
StepHypRef Expression
1 ssuni 3593 . . . . 5 ((xy y B) → x B)
21expcom 109 . . . 4 (y B → (xyx B))
32rexlimiv 2421 . . 3 (y B xyx B)
43ralimi 2378 . 2 (x A y B xyx A x B)
5 unissb 3601 . 2 ( A Bx A x B)
64, 5sylibr 137 1 (x A y B xy A B)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1390  wral 2300  wrex 2301  wss 2911   cuni 3571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572
This theorem is referenced by:  unidif  3603
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