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Theorem iunss2 3693
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 3602. (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 3690 . . 3 (y B 𝐶𝐷𝐶 y B 𝐷)
21ralimi 2378 . 2 (x A y B 𝐶𝐷x A 𝐶 y B 𝐷)
3 iunss 3689 . 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 2300  wrex 2301  wss 2911   ciun 3648
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-iun 3650
This theorem is referenced by:  iunxdif2  3696  rdgss  5910
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