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Theorem unissb 4045
 Description: Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.)
Assertion
Ref Expression
unissb
Distinct variable groups:   ,   ,

Proof of Theorem unissb
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eluni 4018 . . . . . 6
21imbi1i 316 . . . . 5
3 19.23v 1914 . . . . 5
42, 3bitr4i 244 . . . 4
54albii 1575 . . 3
6 alcom 1752 . . . 4
7 19.21v 1913 . . . . . 6
8 impexp 434 . . . . . . . 8
9 bi2.04 351 . . . . . . . 8
108, 9bitri 241 . . . . . . 7
1110albii 1575 . . . . . 6
12 dfss2 3337 . . . . . . 7
1312imbi2i 304 . . . . . 6
147, 11, 133bitr4i 269 . . . . 5
1514albii 1575 . . . 4
166, 15bitri 241 . . 3
175, 16bitri 241 . 2
18 dfss2 3337 . 2
19 df-ral 2710 . 2
2017, 18, 193bitr4i 269 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wal 1549  wex 1550   wcel 1725  wral 2705   wss 3320  cuni 4015 This theorem is referenced by:  uniss2  4046  ssunieq  4048  sspwuni  4176  pwssb  4177  ordunisssuc  4684  bm2.5ii  4786  sorpssuni  6531  sbthlem1  7217  ordunifi  7357  isfinite2  7365  cflim2  8143  fin23lem16  8215  fin23lem29  8221  fin1a2lem11  8290  fin1a2lem13  8292  itunitc  8301  zorng  8384  wuncval2  8622  suplem1pr  8929  suplem2pr  8930  mrcuni  13846  ipodrsfi  14589  mrelatlub  14612  subgint  14964  efgval  15349  toponmre  17157  neips  17177  neiuni  17186  alexsubALTlem2  18079  alexsubALTlem3  18080  tgpconcompeqg  18141  unidmvol  24584  sxbrsigalem0  24621  dya2iocuni  24633  dya2iocucvr  24634  ovoliunnfl  26248  voliunnfl  26250  volsupnfl  26251  topjoin  26394  fnejoin1  26397  fnejoin2  26398  intidl  26639  unichnidl  26641 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-v 2958  df-in 3327  df-ss 3334  df-uni 4016
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