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Theorem unissb 3921
 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 3894 . . . . . 6
21imbi1i 315 . . . . 5
3 19.23v 1891 . . . . 5
42, 3bitr4i 243 . . . 4
54albii 1566 . . 3
6 alcom 1737 . . . 4
7 19.21v 1890 . . . . . 6
8 impexp 433 . . . . . . . 8
9 bi2.04 350 . . . . . . . 8
108, 9bitri 240 . . . . . . 7
1110albii 1566 . . . . . 6
12 dfss2 3262 . . . . . . 7
1312imbi2i 303 . . . . . 6
147, 11, 133bitr4i 268 . . . . 5
1514albii 1566 . . . 4
166, 15bitri 240 . . 3
175, 16bitri 240 . 2
18 dfss2 3262 . 2
19 df-ral 2619 . 2
2017, 18, 193bitr4i 268 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358  wal 1540  wex 1541   wcel 1710  wral 2614   wss 3257  cuni 3891 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-uni 3892 This theorem is referenced by:  uniss2  3922  ssunieq  3924  sspwuni  4051  pwssb  4052
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