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Theorem ssiun2s 4010
 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 (C AD x A B)
Distinct variable groups:   x,A   x,C   x,D
Allowed substitution hint:   B(x)

Proof of Theorem ssiun2s
StepHypRef Expression
1 nfcv 2489 . 2 xC
2 nfcv 2489 . . 3 xD
3 nfiu1 3997 . . 3 xx A B
42, 3nfss 3266 . 2 x D x A B
5 ssiun2s.1 . . 3 (x = CB = D)
65sseq1d 3298 . 2 (x = C → (B x A BD x A B))
7 ssiun2 4009 . 2 (x AB x A B)
81, 4, 6, 7vtoclgaf 2919 1 (C AD x A B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710   ⊆ wss 3257  ∪ciun 3969 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-iun 3971 This theorem is referenced by: (None)
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