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Theorem ssiun 3778
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 3020 . . . . 5 (𝐶𝐵 → (𝑦𝐶𝑦𝐵))
21reximi 2471 . . . 4 (∃𝑥𝐴 𝐶𝐵 → ∃𝑥𝐴 (𝑦𝐶𝑦𝐵))
3 r19.37av 2521 . . . 4 (∃𝑥𝐴 (𝑦𝐶𝑦𝐵) → (𝑦𝐶 → ∃𝑥𝐴 𝑦𝐵))
42, 3syl 14 . . 3 (∃𝑥𝐴 𝐶𝐵 → (𝑦𝐶 → ∃𝑥𝐴 𝑦𝐵))
5 eliun 3740 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
64, 5syl6ibr 161 . 2 (∃𝑥𝐴 𝐶𝐵 → (𝑦𝐶𝑦 𝑥𝐴 𝐵))
76ssrdv 3032 1 (∃𝑥𝐴 𝐶𝐵𝐶 𝑥𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1439  wrex 2361  wss 3000   ciun 3736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-in 3006  df-ss 3013  df-iun 3738
This theorem is referenced by:  iunss2  3781  iunpwss  3826  iunpw  4315
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