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Theorem ssiun2sf 23963
 Description: Subset relationship for an indexed union. (Contributed by Thierry Arnoux, 31-Dec-2016.)
Hypotheses
Ref Expression
ssiun2sf.1
ssiun2sf.2
ssiun2sf.3
ssiun2sf.4
Assertion
Ref Expression
ssiun2sf

Proof of Theorem ssiun2sf
StepHypRef Expression
1 ssiun2sf.2 . . 3
2 ssiun2sf.1 . . . . 5
31, 2nfel 2548 . . . 4
4 ssiun2sf.3 . . . . 5
5 nfiu1 4081 . . . . 5
64, 5nfss 3301 . . . 4
73, 6nfim 1828 . . 3
8 eleq1 2464 . . . 4
9 ssiun2sf.4 . . . . 5
109sseq1d 3335 . . . 4
118, 10imbi12d 312 . . 3
12 ssiun2 4094 . . 3
131, 7, 11, 12vtoclgf 2970 . 2
1413pm2.43i 45 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1649   wcel 1721  wnfc 2527   wss 3280  ciun 4053 This theorem is referenced by:  iundisj2f  23983  voliune  24538  volfiniune  24539 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385 This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-v 2918  df-in 3287  df-ss 3294  df-iun 4055
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