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Theorem tcsni 9161
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 ∈ V
Assertion
Ref Expression
tcsni (TC‘{𝐴}) = ((TC‘𝐴) ∪ {𝐴})

Proof of Theorem tcsni
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 tc2.1 . . . . . 6 𝐴 ∈ V
21snss 4691 . . . . 5 (𝐴𝑥 ↔ {𝐴} ⊆ 𝑥)
32anbi1i 626 . . . 4 ((𝐴𝑥 ∧ Tr 𝑥) ↔ ({𝐴} ⊆ 𝑥 ∧ Tr 𝑥))
43abbii 2886 . . 3 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} = {𝑥 ∣ ({𝐴} ⊆ 𝑥 ∧ Tr 𝑥)}
54inteqi 4853 . 2 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} = {𝑥 ∣ ({𝐴} ⊆ 𝑥 ∧ Tr 𝑥)}
61tc2 9160 . 2 ((TC‘𝐴) ∪ {𝐴}) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
7 snex 5305 . . 3 {𝐴} ∈ V
8 tcvalg 9156 . . 3 ({𝐴} ∈ V → (TC‘{𝐴}) = {𝑥 ∣ ({𝐴} ⊆ 𝑥 ∧ Tr 𝑥)})
97, 8ax-mp 5 . 2 (TC‘{𝐴}) = {𝑥 ∣ ({𝐴} ⊆ 𝑥 ∧ Tr 𝑥)}
105, 6, 93eqtr4ri 2855 1 (TC‘{𝐴}) = ((TC‘𝐴) ∪ {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wcel 2115  {cab 2799  Vcvv 3471  cun 3908  wss 3910  {csn 4540   cint 4849  Tr wtr 5145  cfv 6328  TCctc 9154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-inf2 9080
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-iin 4895  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7556  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-tc 9155
This theorem is referenced by: (None)
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