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Theorem ss2iun 4902
Description: Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ss2iun (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 𝑥𝐴 𝐶)

Proof of Theorem ss2iun
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssel 3911 . . . . 5 (𝐵𝐶 → (𝑦𝐵𝑦𝐶))
21ralimi 3131 . . . 4 (∀𝑥𝐴 𝐵𝐶 → ∀𝑥𝐴 (𝑦𝐵𝑦𝐶))
3 rexim 3207 . . . 4 (∀𝑥𝐴 (𝑦𝐵𝑦𝐶) → (∃𝑥𝐴 𝑦𝐵 → ∃𝑥𝐴 𝑦𝐶))
42, 3syl 17 . . 3 (∀𝑥𝐴 𝐵𝐶 → (∃𝑥𝐴 𝑦𝐵 → ∃𝑥𝐴 𝑦𝐶))
5 eliun 4888 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
6 eliun 4888 . . 3 (𝑦 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑦𝐶)
74, 5, 63imtr4g 299 . 2 (∀𝑥𝐴 𝐵𝐶 → (𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐶))
87ssrdv 3924 1 (∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2112  wral 3109  wrex 3110  wss 3884   ciun 4884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-in 3891  df-ss 3901  df-iun 4886
This theorem is referenced by:  iuneq2  4903  abnexg  7462  oawordri  8163  omwordri  8185  oewordri  8205  oeworde  8206  r1val1  9203  cfslb2n  9683  imasaddvallem  16797  dprdss  19147  tgcmp  22009  txcmplem1  22249  txcmplem2  22250  xkococnlem  22267  alexsubALT  22659  ptcmplem3  22662  metnrmlem2  23468  uniiccvol  24187  dvfval  24503  gsumpart  30743  bnj1145  32373  bnj1136  32377  filnetlem3  33836  poimirlem32  35082  sstotbnd2  35205  equivtotbnd  35209  trclrelexplem  40399  corcltrcl  40427  cotrclrcl  40430  ovolval5lem2  43279  ovolval5lem3  43280  smflimsuplem7  43444
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