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Theorem ssiun2 3725
 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 2385 . . . 4 ((𝑥𝐴𝑦𝐵) → ∃𝑥𝐴 𝑦𝐵)
21ex 112 . . 3 (𝑥𝐴 → (𝑦𝐵 → ∃𝑥𝐴 𝑦𝐵))
3 eliun 3686 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
42, 3syl6ibr 155 . 2 (𝑥𝐴 → (𝑦𝐵𝑦 𝑥𝐴 𝐵))
54ssrdv 2976 1 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1407  ∃wrex 2322   ⊆ wss 2942  ∪ ciun 3682 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036 This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-in 2949  df-ss 2956  df-iun 3684 This theorem is referenced by:  ssiun2s  3726  triun  3892
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