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Theorem ssiun2 5005
Description: Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ssiun2 (𝑥𝐴𝐵 𝑥𝐴 𝐵)

Proof of Theorem ssiun2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 rspe 3230 . . . 4 ((𝑥𝐴𝑦𝐵) → ∃𝑥𝐴 𝑦𝐵)
21ex 413 . . 3 (𝑥𝐴 → (𝑦𝐵 → ∃𝑥𝐴 𝑦𝐵))
3 eliun 4956 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
42, 3syl6ibr 251 . 2 (𝑥𝐴 → (𝑦𝐵𝑦 𝑥𝐴 𝐵))
54ssrdv 3948 1 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wrex 3071  wss 3908   ciun 4952
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rex 3072  df-v 3445  df-in 3915  df-ss 3925  df-iun 4954
This theorem is referenced by:  ssiun2s  5006  disjxiun  5100  triun  5235  iunopeqop  5476  ixpf  8816  ixpiunwdom  9484  r1sdom  9668  r1val1  9680  rankuni2b  9747  rankval4  9761  cplem1  9783  domtriomlem  10336  ac6num  10373  iunfo  10433  iundom2g  10434  pwfseqlem3  10554  inar1  10669  tskuni  10677  iunconnlem  22724  ptclsg  22912  ovoliunlem1  24812  limciun  25204  ssiun2sf  31323  djussxp2  31409  suppovss  31441  bnj906  33370  bnj999  33398  bnj1014  33401  bnj1408  33476  rdgssun  35781  cpcolld  42443  iunmapss  43335  ssmapsn  43336  sge0iunmpt  44554  sge0iun  44555  voliunsge0lem  44608  omeiunltfirp  44655
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