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Theorem iunss2 3672
 Description: A subclass condition on the members of two indexed classes 𝐶(x) and 𝐷(y) that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59. Compare uniss2 3581. (Contributed by NM, 9-Dec-2004.)
Assertion
Ref Expression
iunss2 (x A y B 𝐶𝐷 x A 𝐶 y B 𝐷)
Distinct variable groups:   x,y   x,B   y,𝐶   x,𝐷
Allowed substitution hints:   A(x,y)   B(y)   𝐶(x)   𝐷(y)

Proof of Theorem iunss2
StepHypRef Expression
1 ssiun 3669 . . 3 (y B 𝐶𝐷𝐶 y B 𝐷)
21ralimi 2358 . 2 (x A y B 𝐶𝐷x A 𝐶 y B 𝐷)
3 iunss 3668 . 2 ( x A 𝐶 y B 𝐷x A 𝐶 y B 𝐷)
42, 3sylibr 137 1 (x A y B 𝐶𝐷 x A 𝐶 y B 𝐷)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wral 2280  ∃wrex 2281   ⊆ wss 2890  ∪ ciun 3627 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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-in 2897  df-ss 2904  df-iun 3629 This theorem is referenced by:  iunxdif2  3675  rdgss  5886
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