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Theorem tcsni 8502
 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 4259 . . . . 5 (𝐴𝑥 ↔ {𝐴} ⊆ 𝑥)
32anbi1i 727 . . . 4 ((𝐴𝑥 ∧ Tr 𝑥) ↔ ({𝐴} ⊆ 𝑥 ∧ Tr 𝑥))
43abbii 2726 . . 3 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} = {𝑥 ∣ ({𝐴} ⊆ 𝑥 ∧ Tr 𝑥)}
54inteqi 4414 . 2 {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)} = {𝑥 ∣ ({𝐴} ⊆ 𝑥 ∧ Tr 𝑥)}
61tc2 8501 . 2 ((TC‘𝐴) ∪ {𝐴}) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}
7 snex 4835 . . 3 {𝐴} ∈ V
8 tcvalg 8497 . . 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 2643 1 (TC‘{𝐴}) = ((TC‘𝐴) ∪ {𝐴})
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  {csn 4125  ∩ cint 4410  Tr wtr 4680  ‘cfv 5804  TCctc 8495 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-tc 8496 This theorem is referenced by: (None)
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