MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssiun Structured version   Visualization version   GIF version

Theorem ssiun 4488
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 3557 . . . . 5 (𝐶𝐵 → (𝑦𝐶𝑦𝐵))
21reximi 2989 . . . 4 (∃𝑥𝐴 𝐶𝐵 → ∃𝑥𝐴 (𝑦𝐶𝑦𝐵))
3 r19.37v 3063 . . . 4 (∃𝑥𝐴 (𝑦𝐶𝑦𝐵) → (𝑦𝐶 → ∃𝑥𝐴 𝑦𝐵))
42, 3syl 17 . . 3 (∃𝑥𝐴 𝐶𝐵 → (𝑦𝐶 → ∃𝑥𝐴 𝑦𝐵))
5 eliun 4450 . . 3 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
64, 5syl6ibr 240 . 2 (∃𝑥𝐴 𝐶𝐵 → (𝑦𝐶𝑦 𝑥𝐴 𝐵))
76ssrdv 3569 1 (∃𝑥𝐴 𝐶𝐵𝐶 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1975  wrex 2892  wss 3535   ciun 4445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-v 3170  df-in 3542  df-ss 3549  df-iun 4447
This theorem is referenced by:  iunss2  4491  iunpwss  4541  iunpw  6843  wfrdmcl  7283  onfununi  7298  oen0  7526  trcl  8460  rtrclreclem1  13588  rtrclreclem2  13589  trpredtr  30776  dftrpred3g  30779  frrlem5e  30834
  Copyright terms: Public domain W3C validator