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Theorem tskmid 10454
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 3073 . . 3 𝑥 ∈ Tarski (𝐴𝑥𝐴𝑥)
3 elintrabg 4872 . . 3 (𝐴𝑉 → (𝐴 {𝑥 ∈ Tarski ∣ 𝐴𝑥} ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐴𝑥)))
42, 3mpbiri 261 . 2 (𝐴𝑉𝐴 {𝑥 ∈ Tarski ∣ 𝐴𝑥})
5 tskmval 10453 . 2 (𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
64, 5eleqtrrd 2841 1 (𝐴𝑉𝐴 ∈ (tarskiMap‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2110  wral 3061  {crab 3065   cint 4859  cfv 6380  Tarskictsk 10362  tarskiMapctskm 10451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-groth 10437
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-tsk 10363  df-tskm 10452
This theorem is referenced by:  eltskm  10457
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