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Theorem ssiun2 4529
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 2997 . . . 4 ((𝑥𝐴𝑦𝐵) → ∃𝑥𝐴 𝑦𝐵)
21ex 450 . . 3 (𝑥𝐴 → (𝑦𝐵 → ∃𝑥𝐴 𝑦𝐵))
3 eliun 4490 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
42, 3syl6ibr 242 . 2 (𝑥𝐴 → (𝑦𝐵𝑦 𝑥𝐴 𝐵))
54ssrdv 3589 1 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  wrex 2908  wss 3555   ciun 4485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-in 3562  df-ss 3569  df-iun 4487
This theorem is referenced by:  ssiun2s  4530  disjxiun  4609  disjxiunOLD  4610  triun  4726  iunopeqop  4941  ixpf  7874  ixpiunwdom  8440  r1sdom  8581  r1val1  8593  rankuni2b  8660  rankval4  8674  cplem1  8696  domtriomlem  9208  ac6num  9245  iunfo  9305  iundom2g  9306  pwfseqlem3  9426  inar1  9541  tskuni  9549  iunconnlem  21140  ptclsg  21328  ovoliunlem1  23177  limciun  23564  ssiun2sf  29223  bnj906  30708  bnj999  30735  bnj1014  30738  bnj1408  30812  trpredrec  31439  iunmapss  38881  ssmapsn  38882  sge0iunmpt  39942  sge0iun  39943  voliunsge0lem  39996  omeiunltfirp  40040
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