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Theorem unissb 3573
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  U.  C_  C_
Distinct variable groups:   ,   ,

Proof of Theorem unissb
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eluni 3546 . . . . . 6  U.
21imbi1i 227 . . . . 5  U.
3 19.23v 1736 . . . . 5
42, 3bitr4i 176 . . . 4  U.
54albii 1332 . . 3  U.
6 alcom 1340 . . . 4
7 19.21v 1726 . . . . . 6
8 impexp 250 . . . . . . . 8
9 bi2.04 237 . . . . . . . 8
108, 9bitri 173 . . . . . . 7
1110albii 1332 . . . . . 6
12 dfss2 2902 . . . . . . 7 
C_
1312imbi2i 215 . . . . . 6 
C_
147, 11, 133bitr4i 201 . . . . 5  C_
1514albii 1332 . . . 4  C_
166, 15bitri 173 . . 3  C_
175, 16bitri 173 . 2  U.  C_
18 dfss2 2902 . 2  U.  C_  U.
19 df-ral 2280 . 2  C_ 
C_
2017, 18, 193bitr4i 201 1  U.  C_  C_
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98  wal 1221  wex 1354   wcel 1366  wral 2275    C_ wss 2885   U.cuni 3543
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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-v 2528  df-in 2892  df-ss 2899  df-uni 3544
This theorem is referenced by:  uniss2  3574  ssunieq  3576  sspwuni  3702  pwssb  3703  bm2.5ii  4160
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