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Theorem rabsnif 4032
 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:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem rabsnif
StepHypRef Expression
1 rabsnifsb 4031 . . 3
2 rabsnif.f . . . . 5
32sbcieg 3288 . . . 4
43ifbid 3894 . . 3
51, 4syl5eq 2517 . 2
6 rab0 3756 . . . 4
7 ifid 3909 . . . 4
86, 7eqtr4i 2496 . . 3
9 snprc 4027 . . . . 5
109biimpi 199 . . . 4
11 rabeq 3024 . . . 4
1210, 11syl 17 . . 3
1310ifeq1d 3890 . . 3
148, 12, 133eqtr4a 2531 . 2
155, 14pm2.61i 169 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 189   wceq 1452   wcel 1904  crab 2760  cvv 3031  wsbc 3255  c0 3722  cif 3872  csn 3959 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-nul 3723  df-if 3873  df-sn 3960 This theorem is referenced by:  suppsnop  6947  m1detdiag  19699  1loopgrvd2  39725  1hevtxdg1  39728  1egrvtxdg1  39732
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