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Theorem ssiun2 5047
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 3249 . . . 4 ((𝑥𝐴𝑦𝐵) → ∃𝑥𝐴 𝑦𝐵)
21ex 412 . . 3 (𝑥𝐴 → (𝑦𝐵 → ∃𝑥𝐴 𝑦𝐵))
3 eliun 4995 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
42, 3imbitrrdi 252 . 2 (𝑥𝐴 → (𝑦𝐵𝑦 𝑥𝐴 𝐵))
54ssrdv 3989 1 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wrex 3070  wss 3951   ciun 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rex 3071  df-v 3482  df-ss 3968  df-iun 4993
This theorem is referenced by:  ssiun2s  5048  disjxiun  5140  triun  5274  iunopeqop  5526  ixpf  8960  ixpiunwdom  9630  r1sdom  9814  r1val1  9826  rankuni2b  9893  rankval4  9907  cplem1  9929  domtriomlem  10482  ac6num  10519  iunfo  10579  iundom2g  10580  pwfseqlem3  10700  inar1  10815  tskuni  10823  iunconnlem  23435  ptclsg  23623  ovoliunlem1  25537  limciun  25929  ssiun2sf  32572  iunxpssiun1  32581  djussxp2  32658  suppovss  32690  bnj906  34944  bnj999  34972  bnj1014  34975  bnj1408  35050  rdgssun  37379  cpcolld  44277  iunmapss  45220  ssmapsn  45221  sge0iunmpt  46433  sge0iun  46434  voliunsge0lem  46487  omeiunltfirp  46534
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