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Theorem rabrsndc 3627
 Description: A class abstraction over a decidable proposition restricted to a singleton is either the empty set or the singleton itself. (Contributed by Jim Kingdon, 8-Aug-2018.)
Hypotheses
Ref Expression
rabrsndc.1
rabrsndc.2 DECID
Assertion
Ref Expression
rabrsndc
Distinct variable group:   ,
Allowed substitution hints:   ()   ()

Proof of Theorem rabrsndc
StepHypRef Expression
1 rabrsndc.1 . . . . . 6
2 rabrsndc.2 . . . . . . . 8 DECID
3 pm2.1dc 823 . . . . . . . 8 DECID
42, 3ax-mp 5 . . . . . . 7
54sbcth 2950 . . . . . 6
61, 5ax-mp 5 . . . . 5
7 sbcor 2981 . . . . 5
86, 7mpbi 144 . . . 4
9 ralsns 3597 . . . . . 6
101, 9ax-mp 5 . . . . 5
11 ralsns 3597 . . . . . 6
121, 11ax-mp 5 . . . . 5
1310, 12orbi12i 754 . . . 4
148, 13mpbir 145 . . 3
15 rabeq0 3423 . . . 4
16 eqcom 2159 . . . . 5
17 rabid2 2633 . . . . 5
1816, 17bitri 183 . . . 4
1915, 18orbi12i 754 . . 3
2014, 19mpbir 145 . 2
21 eqeq1 2164 . . 3
22 eqeq1 2164 . . 3
2321, 22orbi12d 783 . 2
2420, 23mpbiri 167 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 104   wo 698  DECID wdc 820   wceq 1335   wcel 2128  wral 2435  crab 2439  cvv 2712  wsbc 2937  c0 3394  csn 3560 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-nul 3395  df-sn 3566 This theorem is referenced by: (None)
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