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Theorem ssiun2s 3692
 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 2175 . 2 x𝐶
2 nfcv 2175 . . 3 x𝐷
3 nfiu1 3678 . . 3 x x A B
42, 3nfss 2932 . 2 x 𝐷 x A B
5 ssiun2s.1 . . 3 (x = 𝐶B = 𝐷)
65sseq1d 2966 . 2 (x = 𝐶 → (B x A B𝐷 x A B))
7 ssiun2 3691 . 2 (x AB x A B)
81, 4, 6, 7vtoclgaf 2612 1 (𝐶 A𝐷 x A B)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390   ⊆ wss 2911  ∪ ciun 3648 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-iun 3650 This theorem is referenced by: (None)
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