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Theorem ssiun2s 5052
Description: Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.)
Hypothesis
Ref Expression
ssiun2s.1 (𝑥 = 𝐶𝐵 = 𝐷)
Assertion
Ref Expression
ssiun2s (𝐶𝐴𝐷 𝑥𝐴 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem ssiun2s
StepHypRef Expression
1 nfcv 2904 . 2 𝑥𝐶
2 nfcv 2904 . . 3 𝑥𝐷
3 nfiu1 5032 . . 3 𝑥 𝑥𝐴 𝐵
42, 3nfss 3975 . 2 𝑥 𝐷 𝑥𝐴 𝐵
5 ssiun2s.1 . . 3 (𝑥 = 𝐶𝐵 = 𝐷)
65sseq1d 4014 . 2 (𝑥 = 𝐶 → (𝐵 𝑥𝐴 𝐵𝐷 𝑥𝐴 𝐵))
7 ssiun2 5051 . 2 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
81, 4, 6, 7vtoclgaf 3565 1 (𝐶𝐴𝐷 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wss 3949   ciun 4998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-v 3477  df-in 3956  df-ss 3966  df-iun 5000
This theorem is referenced by:  fviunfun  7931  onfununi  8341  oaordi  8546  omordi  8566  dffi3  9426  alephordi  10069  domtriomlem  10437  pwxpndom2  10660  wunex2  10733  imasaddvallem  17475  imasvscaval  17484  iundisj2  25066  voliunlem1  25067  volsup  25073  iundisj2fi  32008  bnj906  33941  bnj1137  34006  bnj1408  34047  cvmliftlem10  34285  cvmliftlem13  34287  sstotbnd2  36642  mapdrvallem3  40517  onsucunifi  42120  fvmptiunrelexplb0d  42435  fvmptiunrelexplb1d  42437  corclrcl  42458  trclrelexplem  42462  corcltrcl  42490  cotrclrcl  42493  iunincfi  43783  iundjiunlem  45175  meaiuninc3v  45200  caratheodorylem1  45242  ovnhoilem1  45317
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