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Theorem rabrsn 4102
 Description: A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by Alexander van der Vekens, 22-Dec-2017.) (Proof shortened by AV, 21-Jul-2019.)
Assertion
Ref Expression
rabrsn
Distinct variable group:   ,
Allowed substitution hints:   ()   ()

Proof of Theorem rabrsn
StepHypRef Expression
1 rabsnifsb 4100 . . 3
21eqeq2i 2475 . 2
3 ifeqor 3988 . . . 4
4 orcom 387 . . . 4
53, 4mpbir 209 . . 3
6 eqeq1 2461 . . . 4
7 eqeq1 2461 . . . 4
86, 7orbi12d 709 . . 3
95, 8mpbiri 233 . 2
102, 9sylbi 195 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wo 368   wceq 1395  crab 2811  wsbc 3327  c0 3793  cif 3944  csn 4032 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-nul 3794  df-if 3945  df-sn 4033 This theorem is referenced by:  hashrabrsn  12442  hashrabsn01  12443  hashrabsn1  12444  dvnprodlem3  31906
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