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Theorem tcsni 8170
Description: The transitive closure of a singleton. Proof suggested by Gérard Lang. (Contributed by Mario Carneiro, 4-Jun-2015.)
Hypothesis
Ref Expression
tc2.1  |-  A  e. 
_V
Assertion
Ref Expression
tcsni  |-  ( TC
`  { A }
)  =  ( ( TC `  A )  u.  { A }
)

Proof of Theorem tcsni
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 tc2.1 . . . . . 6  |-  A  e. 
_V
21snss 4151 . . . . 5  |-  ( A  e.  x  <->  { A }  C_  x )
32anbi1i 695 . . . 4  |-  ( ( A  e.  x  /\  Tr  x )  <->  ( { A }  C_  x  /\  Tr  x ) )
43abbii 2601 . . 3  |-  { x  |  ( A  e.  x  /\  Tr  x
) }  =  {
x  |  ( { A }  C_  x  /\  Tr  x ) }
54inteqi 4286 . 2  |-  |^| { x  |  ( A  e.  x  /\  Tr  x
) }  =  |^| { x  |  ( { A }  C_  x  /\  Tr  x ) }
61tc2 8169 . 2  |-  ( ( TC `  A )  u.  { A }
)  =  |^| { x  |  ( A  e.  x  /\  Tr  x
) }
7 snex 4688 . . 3  |-  { A }  e.  _V
8 tcvalg 8165 . . 3  |-  ( { A }  e.  _V  ->  ( TC `  { A } )  =  |^| { x  |  ( { A }  C_  x  /\  Tr  x ) } )
97, 8ax-mp 5 . 2  |-  ( TC
`  { A }
)  =  |^| { x  |  ( { A }  C_  x  /\  Tr  x ) }
105, 6, 93eqtr4ri 2507 1  |-  ( TC
`  { A }
)  =  ( ( TC `  A )  u.  { A }
)
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   _Vcvv 3113    u. cun 3474    C_ wss 3476   {csn 4027   |^|cint 4282   Tr wtr 4540   ` cfv 5586   TCctc 8163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-om 6679  df-recs 7039  df-rdg 7073  df-tc 8164
This theorem is referenced by: (None)
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