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Theorem tskmid 10880
Description: The set 𝐴 is an element of the smallest Tarski class that contains 𝐴. CLASSES1 th. 5. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
Assertion
Ref Expression
tskmid (𝐴𝑉𝐴 ∈ (tarskiMap‘𝐴))

Proof of Theorem tskmid
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . 4 (𝐴𝑥𝐴𝑥)
21rgenw 3065 . . 3 𝑥 ∈ Tarski (𝐴𝑥𝐴𝑥)
3 elintrabg 4961 . . 3 (𝐴𝑉 → (𝐴 {𝑥 ∈ Tarski ∣ 𝐴𝑥} ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐴𝑥)))
42, 3mpbiri 258 . 2 (𝐴𝑉𝐴 {𝑥 ∈ Tarski ∣ 𝐴𝑥})
5 tskmval 10879 . 2 (𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
64, 5eleqtrrd 2844 1 (𝐴𝑉𝐴 ∈ (tarskiMap‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wral 3061  {crab 3436   cint 4946  cfv 6561  Tarskictsk 10788  tarskiMapctskm 10877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-groth 10863
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-tsk 10789  df-tskm 10878
This theorem is referenced by:  eltskm  10883
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