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Theorem ssiun2sf 30239
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 2989 . . . 4 𝑥 𝐶𝐴
4 ssiun2sf.3 . . . . 5 𝑥𝐷
5 nfiu1 4944 . . . . 5 𝑥 𝑥𝐴 𝐵
64, 5nfss 3957 . . . 4 𝑥 𝐷 𝑥𝐴 𝐵
73, 6nfim 1888 . . 3 𝑥(𝐶𝐴𝐷 𝑥𝐴 𝐵)
8 eleq1 2897 . . . 4 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
9 ssiun2sf.4 . . . . 5 (𝑥 = 𝐶𝐵 = 𝐷)
109sseq1d 3995 . . . 4 (𝑥 = 𝐶 → (𝐵 𝑥𝐴 𝐵𝐷 𝑥𝐴 𝐵))
118, 10imbi12d 346 . . 3 (𝑥 = 𝐶 → ((𝑥𝐴𝐵 𝑥𝐴 𝐵) ↔ (𝐶𝐴𝐷 𝑥𝐴 𝐵)))
12 ssiun2 4962 . . 3 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
131, 7, 11, 12vtoclgf 3563 . 2 (𝐶𝐴 → (𝐶𝐴𝐷 𝑥𝐴 𝐵))
1413pm2.43i 52 1 (𝐶𝐴𝐷 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wnfc 2958  wss 3933   ciun 4910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-v 3494  df-in 3940  df-ss 3949  df-iun 4912
This theorem is referenced by:  iundisj2f  30268  esum2dlem  31250  voliune  31387  volfiniune  31388
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