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Theorem tc00 8213
Description: The transitive closure is empty iff its argument is. Proof suggested by Gérard Lang. (Contributed by Mario Carneiro, 4-Jun-2015.)
Assertion
Ref Expression
tc00  |-  ( A  e.  V  ->  (
( TC `  A
)  =  (/)  <->  A  =  (/) ) )

Proof of Theorem tc00
StepHypRef Expression
1 tcid 8204 . . 3  |-  ( A  e.  V  ->  A  C_  ( TC `  A
) )
2 sseq0 3773 . . . 4  |-  ( ( A  C_  ( TC `  A )  /\  ( TC `  A )  =  (/) )  ->  A  =  (/) )
32ex 434 . . 3  |-  ( A 
C_  ( TC `  A )  ->  (
( TC `  A
)  =  (/)  ->  A  =  (/) ) )
41, 3syl 17 . 2  |-  ( A  e.  V  ->  (
( TC `  A
)  =  (/)  ->  A  =  (/) ) )
5 fveq2 5851 . . 3  |-  ( A  =  (/)  ->  ( TC
`  A )  =  ( TC `  (/) ) )
6 tc0 8212 . . 3  |-  ( TC
`  (/) )  =  (/)
75, 6syl6eq 2461 . 2  |-  ( A  =  (/)  ->  ( TC
`  A )  =  (/) )
84, 7impbid1 205 1  |-  ( A  e.  V  ->  (
( TC `  A
)  =  (/)  <->  A  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    = wceq 1407    e. wcel 1844    C_ wss 3416   (/)c0 3740   ` cfv 5571   TCctc 8201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-om 6686  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-tc 8202
This theorem is referenced by: (None)
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