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Theorem rabsnif 4061
Description: A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 21-Jul-2019.)
Hypothesis
Ref Expression
rabsnif.f
Assertion
Ref Expression
rabsnif
Distinct variable groups:   ,   ,

Proof of Theorem rabsnif
StepHypRef Expression
1 rabsnifsb 4060 . . 3
2 rabsnif.f . . . . 5
32sbcieg 3330 . . . 4
43ifbid 3927 . . 3
51, 4syl5eq 2507 . 2
6 rab0 3772 . . . 4
7 ifid 3942 . . . 4
86, 7eqtr4i 2486 . . 3
9 snprc 4056 . . . . 5
109biimpi 194 . . . 4
11 rabeq 3075 . . . 4
1210, 11syl 16 . . 3
1310ifeq1d 3923 . . 3
148, 12, 133eqtr4a 2521 . 2
155, 14pm2.61i 164 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  =wceq 1370  e.wcel 1758  {crab 2804   cvv 3081  [.wsbc 3297   c0 3751  ifcif 3905  {csn 3993
This theorem is referenced by:  suppsnop  6838  m1detdiag  18671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-rab 2809  df-v 3083  df-sbc 3298  df-dif 3445  df-un 3447  df-nul 3752  df-if 3906  df-sn 3994
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