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Theorem unissb 3638
 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 3611 . . . . . 6
21imbi1i 231 . . . . 5
3 19.23v 1779 . . . . 5
42, 3bitr4i 180 . . . 4
54albii 1375 . . 3
6 alcom 1383 . . . 4
7 19.21v 1769 . . . . . 6
8 impexp 254 . . . . . . . 8
9 bi2.04 241 . . . . . . . 8
108, 9bitri 177 . . . . . . 7
1110albii 1375 . . . . . 6
12 dfss2 2962 . . . . . . 7
1312imbi2i 219 . . . . . 6
147, 11, 133bitr4i 205 . . . . 5
1514albii 1375 . . . 4
166, 15bitri 177 . . 3
175, 16bitri 177 . 2
18 dfss2 2962 . 2
19 df-ral 2328 . 2
2017, 18, 193bitr4i 205 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 101   wb 102  wal 1257  wex 1397   wcel 1409  wral 2323   wss 2945  cuni 3608 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-in 2952  df-ss 2959  df-uni 3609 This theorem is referenced by:  uniss2  3639  ssunieq  3641  sspwuni  3767  pwssb  3768  bm2.5ii  4250
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