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Theorem ssiun 5046
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 3977 . . . . 5 (𝐶𝐵 → (𝑦𝐶𝑦𝐵))
21reximi 3084 . . . 4 (∃𝑥𝐴 𝐶𝐵 → ∃𝑥𝐴 (𝑦𝐶𝑦𝐵))
3 r19.37v 3182 . . . 4 (∃𝑥𝐴 (𝑦𝐶𝑦𝐵) → (𝑦𝐶 → ∃𝑥𝐴 𝑦𝐵))
42, 3syl 17 . . 3 (∃𝑥𝐴 𝐶𝐵 → (𝑦𝐶 → ∃𝑥𝐴 𝑦𝐵))
5 eliun 4995 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
64, 5imbitrrdi 252 . 2 (∃𝑥𝐴 𝐶𝐵 → (𝑦𝐶𝑦 𝑥𝐴 𝐵))
76ssrdv 3989 1 (∃𝑥𝐴 𝐶𝐵𝐶 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-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-ral 3062  df-rex 3071  df-v 3482  df-ss 3968  df-iun 4993
This theorem is referenced by:  iunss2  5049  iunpwss  5107  iunpw  7791  wfrdmclOLD  8357  onfununi  8381  oen0  8624  trcl  9768  rtrclreclem1  15096  rtrclreclem2  15098  constrmon  33785  oacl2g  43343  omcl2  43346  ofoaf  43368
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