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Theorem rabsnifsb 4098
Description: A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 21-Jul-2019.)
Assertion
Ref Expression
rabsnifsb
Distinct variable group:   ,

Proof of Theorem rabsnifsb
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elsni 4054 . . . . . . . 8
2 sbceq1a 3338 . . . . . . . . 9
32biimpd 207 . . . . . . . 8
41, 3syl 16 . . . . . . 7
54imdistani 690 . . . . . 6
65orcd 392 . . . . 5
72bicomd 201 . . . . . . . . 9
81, 7syl 16 . . . . . . . 8
98biimpd 207 . . . . . . 7
109imdistani 690 . . . . . 6
11 noel 3788 . . . . . . . 8
1211pm2.21i 131 . . . . . . 7
1312adantr 465 . . . . . 6
1410, 13jaoi 379 . . . . 5
156, 14impbii 188 . . . 4
1615abbii 2591 . . 3
17 nfv 1707 . . . 4
18 nfv 1707 . . . . . 6
19 nfsbc1v 3347 . . . . . 6
2018, 19nfan 1928 . . . . 5
21 nfv 1707 . . . . . 6
2219nfn 1901 . . . . . 6
2321, 22nfan 1928 . . . . 5
2420, 23nfor 1935 . . . 4
25 eleq1 2529 . . . . . 6
2625anbi1d 704 . . . . 5
27 eleq1 2529 . . . . . 6
2827anbi1d 704 . . . . 5
2926, 28orbi12d 709 . . . 4
3017, 24, 29cbvab 2598 . . 3
3116, 30eqtri 2486 . 2
32 df-rab 2816 . 2
33 df-if 3942 . 2
3431, 32, 333eqtr4i 2496 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  {cab 2442  {crab 2811  [.wsbc 3327   c0 3784  ifcif 3941  {csn 4029
This theorem is referenced by:  rabsnif  4099  rabrsn  4100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-nul 3785  df-if 3942  df-sn 4030
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