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Theorem ssint 3942
 Description: Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.)
Assertion
Ref Expression
ssint
Distinct variable groups:   ,   ,

Proof of Theorem ssint
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfss3 3263 . 2
2 vex 2862 . . . 4
32elint2 3933 . . 3
43ralbii 2638 . 2
5 ralcom 2771 . . 3
6 dfss3 3263 . . . 4
76ralbii 2638 . . 3
85, 7bitr4i 243 . 2
91, 4, 83bitri 262 1
 Colors of variables: wff setvar class Syntax hints:   wb 176   wcel 1710  wral 2614   wss 3257  cint 3926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-int 3927 This theorem is referenced by:  ssintab  3943  ssintub  3944  iinpw  4054  fint  5245
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