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Theorem ssiun2 4715
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 3141 . . . 4 ((𝑥𝐴𝑦𝐵) → ∃𝑥𝐴 𝑦𝐵)
21ex 449 . . 3 (𝑥𝐴 → (𝑦𝐵 → ∃𝑥𝐴 𝑦𝐵))
3 eliun 4676 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
42, 3syl6ibr 242 . 2 (𝑥𝐴 → (𝑦𝐵𝑦 𝑥𝐴 𝐵))
54ssrdv 3750 1 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2139  wrex 3051  wss 3715   ciun 4672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-in 3722  df-ss 3729  df-iun 4674
This theorem is referenced by:  ssiun2s  4716  disjxiun  4801  triun  4918  iunopeqop  5131  ixpf  8096  ixpiunwdom  8661  r1sdom  8810  r1val1  8822  rankuni2b  8889  rankval4  8903  cplem1  8925  domtriomlem  9456  ac6num  9493  iunfo  9553  iundom2g  9554  pwfseqlem3  9674  inar1  9789  tskuni  9797  iunconnlem  21432  ptclsg  21620  ovoliunlem1  23470  limciun  23857  ssiun2sf  29685  bnj906  31307  bnj999  31334  bnj1014  31337  bnj1408  31411  trpredrec  32043  iunmapss  39906  ssmapsn  39907  sge0iunmpt  41138  sge0iun  41139  voliunsge0lem  41192  omeiunltfirp  41239
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