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Theorem tcsni 9735
Description: The transitive closure of a singleton. Proof suggested by GΓ©rard Lang. (Contributed by Mario Carneiro, 4-Jun-2015.)
Hypothesis
Ref Expression
tc2.1 𝐴 ∈ V
Assertion
Ref Expression
tcsni (TCβ€˜{𝐴}) = ((TCβ€˜π΄) βˆͺ {𝐴})

Proof of Theorem tcsni
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 tc2.1 . . . . . 6 𝐴 ∈ V
21snss 4782 . . . . 5 (𝐴 ∈ π‘₯ ↔ {𝐴} βŠ† π‘₯)
32anbi1i 623 . . . 4 ((𝐴 ∈ π‘₯ ∧ Tr π‘₯) ↔ ({𝐴} βŠ† π‘₯ ∧ Tr π‘₯))
43abbii 2794 . . 3 {π‘₯ ∣ (𝐴 ∈ π‘₯ ∧ Tr π‘₯)} = {π‘₯ ∣ ({𝐴} βŠ† π‘₯ ∧ Tr π‘₯)}
54inteqi 4945 . 2 ∩ {π‘₯ ∣ (𝐴 ∈ π‘₯ ∧ Tr π‘₯)} = ∩ {π‘₯ ∣ ({𝐴} βŠ† π‘₯ ∧ Tr π‘₯)}
61tc2 9734 . 2 ((TCβ€˜π΄) βˆͺ {𝐴}) = ∩ {π‘₯ ∣ (𝐴 ∈ π‘₯ ∧ Tr π‘₯)}
7 snex 5422 . . 3 {𝐴} ∈ V
8 tcvalg 9730 . . 3 ({𝐴} ∈ V β†’ (TCβ€˜{𝐴}) = ∩ {π‘₯ ∣ ({𝐴} βŠ† π‘₯ ∧ Tr π‘₯)})
97, 8ax-mp 5 . 2 (TCβ€˜{𝐴}) = ∩ {π‘₯ ∣ ({𝐴} βŠ† π‘₯ ∧ Tr π‘₯)}
105, 6, 93eqtr4ri 2763 1 (TCβ€˜{𝐴}) = ((TCβ€˜π΄) βˆͺ {𝐴})
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {cab 2701  Vcvv 3466   βˆͺ cun 3939   βŠ† wss 3941  {csn 4621  βˆ© cint 4941  Tr wtr 5256  β€˜cfv 6534  TCctc 9728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719  ax-inf2 9633
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-iin 4991  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-om 7850  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-tc 9729
This theorem is referenced by: (None)
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