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Theorem ssiun 5050
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 3988 . . . . 5 (𝐶𝐵 → (𝑦𝐶𝑦𝐵))
21reximi 3081 . . . 4 (∃𝑥𝐴 𝐶𝐵 → ∃𝑥𝐴 (𝑦𝐶𝑦𝐵))
3 r19.37v 3179 . . . 4 (∃𝑥𝐴 (𝑦𝐶𝑦𝐵) → (𝑦𝐶 → ∃𝑥𝐴 𝑦𝐵))
42, 3syl 17 . . 3 (∃𝑥𝐴 𝐶𝐵 → (𝑦𝐶 → ∃𝑥𝐴 𝑦𝐵))
5 eliun 4999 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
64, 5imbitrrdi 252 . 2 (∃𝑥𝐴 𝐶𝐵 → (𝑦𝐶𝑦 𝑥𝐴 𝐵))
76ssrdv 4000 1 (∃𝑥𝐴 𝐶𝐵𝐶 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  wrex 3067  wss 3962   ciun 4995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-v 3479  df-ss 3979  df-iun 4997
This theorem is referenced by:  iunss2  5053  iunpwss  5111  iunpw  7789  wfrdmclOLD  8355  onfununi  8379  oen0  8622  trcl  9765  rtrclreclem1  15092  rtrclreclem2  15094  constrmon  33748  oacl2g  43319  omcl2  43322  ofoaf  43344
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