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Theorem unissb 3761
 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 3734 . . . . . 6
21imbi1i 237 . . . . 5
3 19.23v 1855 . . . . 5
42, 3bitr4i 186 . . . 4
54albii 1446 . . 3
6 alcom 1454 . . . 4
7 19.21v 1845 . . . . . 6
8 impexp 261 . . . . . . . 8
9 bi2.04 247 . . . . . . . 8
108, 9bitri 183 . . . . . . 7
1110albii 1446 . . . . . 6
12 dfss2 3081 . . . . . . 7
1312imbi2i 225 . . . . . 6
147, 11, 133bitr4i 211 . . . . 5
1514albii 1446 . . . 4
166, 15bitri 183 . . 3
175, 16bitri 183 . 2
18 dfss2 3081 . 2
19 df-ral 2419 . 2
2017, 18, 193bitr4i 211 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104  wal 1329  wex 1468   wcel 1480  wral 2414   wss 3066  cuni 3731 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-in 3072  df-ss 3079  df-uni 3732 This theorem is referenced by:  uniss2  3762  ssunieq  3764  sspwuni  3892  pwssb  3893  bm2.5ii  4407  sbthlem1  6838  neipsm  12312  neiuni  12319
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