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Theorem brsingle 25764
 Description: The binary relationship form of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brsingle.1
brsingle.2
Assertion
Ref Expression
brsingle Singleton

Proof of Theorem brsingle
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 brsingle.1 . 2
2 brsingle.2 . 2
3 df-singleton 25708 . 2 Singleton (++)
4 brxp 4911 . . 3
51, 2, 4mpbir2an 888 . 2
6 elsn 3831 . . 3
71ideq 5027 . . 3
86, 7bitr4i 245 . 2
91, 2, 3, 5, 8brtxpsd3 25743 1 Singleton
 Colors of variables: wff set class Syntax hints:   wb 178   wceq 1653   wcel 1726  cvv 2958  csn 3816   class class class wbr 4214   cid 4495   cxp 4878  Singletoncsingle 25684 This theorem is referenced by:  elsingles  25765  fnsingle  25766  fvsingle  25767  brapply  25785  brsuccf  25788  funpartlem  25789 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-eprel 4496  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fo 5462  df-fv 5464  df-1st 6351  df-2nd 6352  df-symdif 25665  df-txp 25700  df-singleton 25708
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