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Theorem tskmid 10250
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 3147 . . 3 𝑥 ∈ Tarski (𝐴𝑥𝐴𝑥)
3 elintrabg 4880 . . 3 (𝐴𝑉 → (𝐴 {𝑥 ∈ Tarski ∣ 𝐴𝑥} ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐴𝑥)))
42, 3mpbiri 259 . 2 (𝐴𝑉𝐴 {𝑥 ∈ Tarski ∣ 𝐴𝑥})
5 tskmval 10249 . 2 (𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
64, 5eleqtrrd 2913 1 (𝐴𝑉𝐴 ∈ (tarskiMap‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  wral 3135  {crab 3139   cint 4867  cfv 6348  Tarskictsk 10158  tarskiMapctskm 10247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-groth 10233
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-tsk 10159  df-tskm 10248
This theorem is referenced by:  eltskm  10253
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