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Theorem difin0ss 3686
 Description: Difference, intersection, and subclass relationship. (Contributed by NM, 30-Apr-1994.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
Assertion
Ref Expression
difin0ss

Proof of Theorem difin0ss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eq0 3634 . 2
2 iman 414 . . . . . 6
3 elin 3522 . . . . . . . 8
4 eldif 3322 . . . . . . . . 9
54anbi1i 677 . . . . . . . 8
63, 5bitri 241 . . . . . . 7
7 ancom 438 . . . . . . 7
8 annim 415 . . . . . . . 8
98anbi2i 676 . . . . . . 7
106, 7, 93bitr2i 265 . . . . . 6
112, 10xchbinxr 303 . . . . 5
12 ax-2 6 . . . . 5
1311, 12sylbir 205 . . . 4
1413al2imi 1570 . . 3
15 dfss2 3329 . . 3
16 dfss2 3329 . . 3
1714, 15, 163imtr4g 262 . 2
181, 17sylbi 188 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 359  wal 1549   wceq 1652   wcel 1725   cdif 3309   cin 3311   wss 3312  c0 3620 This theorem is referenced by:  tz7.7  4599  tfi  4825  lebnumlem3  18978 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-nul 3621
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