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Theorem ssiun2sf 32572
Description: Subset relationship for an indexed union. (Contributed by Thierry Arnoux, 31-Dec-2016.)
Hypotheses
Ref Expression
ssiun2sf.1 𝑥𝐴
ssiun2sf.2 𝑥𝐶
ssiun2sf.3 𝑥𝐷
ssiun2sf.4 (𝑥 = 𝐶𝐵 = 𝐷)
Assertion
Ref Expression
ssiun2sf (𝐶𝐴𝐷 𝑥𝐴 𝐵)

Proof of Theorem ssiun2sf
StepHypRef Expression
1 ssiun2sf.2 . . 3 𝑥𝐶
2 ssiun2sf.1 . . . . 5 𝑥𝐴
31, 2nfel 2920 . . . 4 𝑥 𝐶𝐴
4 ssiun2sf.3 . . . . 5 𝑥𝐷
5 nfiu1 5027 . . . . 5 𝑥 𝑥𝐴 𝐵
64, 5nfss 3976 . . . 4 𝑥 𝐷 𝑥𝐴 𝐵
73, 6nfim 1896 . . 3 𝑥(𝐶𝐴𝐷 𝑥𝐴 𝐵)
8 eleq1 2829 . . . 4 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
9 ssiun2sf.4 . . . . 5 (𝑥 = 𝐶𝐵 = 𝐷)
109sseq1d 4015 . . . 4 (𝑥 = 𝐶 → (𝐵 𝑥𝐴 𝐵𝐷 𝑥𝐴 𝐵))
118, 10imbi12d 344 . . 3 (𝑥 = 𝐶 → ((𝑥𝐴𝐵 𝑥𝐴 𝐵) ↔ (𝐶𝐴𝐷 𝑥𝐴 𝐵)))
12 ssiun2 5047 . . 3 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
131, 7, 11, 12vtoclgf 3569 . 2 (𝐶𝐴 → (𝐶𝐴𝐷 𝑥𝐴 𝐵))
1413pm2.43i 52 1 (𝐶𝐴𝐷 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wnfc 2890  wss 3951   ciun 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-v 3482  df-ss 3968  df-iun 4993
This theorem is referenced by:  iundisj2f  32603  esum2dlem  34093  voliune  34230  volfiniune  34231
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