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Theorem ssiun 4715
 Description: Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ssiun (∃𝑥𝐴 𝐶𝐵𝐶 𝑥𝐴 𝐵)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem ssiun
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssel 3739 . . . . 5 (𝐶𝐵 → (𝑦𝐶𝑦𝐵))
21reximi 3150 . . . 4 (∃𝑥𝐴 𝐶𝐵 → ∃𝑥𝐴 (𝑦𝐶𝑦𝐵))
3 r19.37v 3226 . . . 4 (∃𝑥𝐴 (𝑦𝐶𝑦𝐵) → (𝑦𝐶 → ∃𝑥𝐴 𝑦𝐵))
42, 3syl 17 . . 3 (∃𝑥𝐴 𝐶𝐵 → (𝑦𝐶 → ∃𝑥𝐴 𝑦𝐵))
5 eliun 4677 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
64, 5syl6ibr 242 . 2 (∃𝑥𝐴 𝐶𝐵 → (𝑦𝐶𝑦 𝑥𝐴 𝐵))
76ssrdv 3751 1 (∃𝑥𝐴 𝐶𝐵𝐶 𝑥𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2140  ∃wrex 3052   ⊆ wss 3716  ∪ ciun 4673 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 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741 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 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-v 3343  df-in 3723  df-ss 3730  df-iun 4675 This theorem is referenced by:  iunss2  4718  iunpwss  4771  iunpw  7145  wfrdmcl  7594  onfununi  7609  oen0  7838  trcl  8780  rtrclreclem1  14018  rtrclreclem2  14019  trpredtr  32057  dftrpred3g  32060  frrlem5e  32116
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