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Theorem tcsni 7575
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 3841 . . . . 5  |-  ( A  e.  x  <->  { A }  C_  x )
32anbi1i 676 . . . 4  |-  ( ( A  e.  x  /\  Tr  x )  <->  ( { A }  C_  x  /\  Tr  x ) )
43abbii 2478 . . 3  |-  { x  |  ( A  e.  x  /\  Tr  x
) }  =  {
x  |  ( { A }  C_  x  /\  Tr  x ) }
54inteqi 3968 . 2  |-  |^| { x  |  ( A  e.  x  /\  Tr  x
) }  =  |^| { x  |  ( { A }  C_  x  /\  Tr  x ) }
61tc2 7574 . 2  |-  ( ( TC `  A )  u.  { A }
)  =  |^| { x  |  ( A  e.  x  /\  Tr  x
) }
7 snex 4318 . . 3  |-  { A }  e.  _V
8 tcvalg 7570 . . 3  |-  ( { A }  e.  _V  ->  ( TC `  { A } )  =  |^| { x  |  ( { A }  C_  x  /\  Tr  x ) } )
97, 8ax-mp 8 . 2  |-  ( TC
`  { A }
)  =  |^| { x  |  ( { A }  C_  x  /\  Tr  x ) }
105, 6, 93eqtr4ri 2397 1  |-  ( TC
`  { A }
)  =  ( ( TC `  A )  u.  { A }
)
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1647    e. wcel 1715   {cab 2352   _Vcvv 2873    u. cun 3236    C_ wss 3238   {csn 3729   |^|cint 3964   Tr wtr 4215   ` cfv 5358   TCctc 7568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-recs 6530  df-rdg 6565  df-tc 7569
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