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Theorem tskmid 9775
 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 3026 . . 3 𝑥 ∈ Tarski (𝐴𝑥𝐴𝑥)
3 elintrabg 4597 . . 3 (𝐴𝑉 → (𝐴 {𝑥 ∈ Tarski ∣ 𝐴𝑥} ↔ ∀𝑥 ∈ Tarski (𝐴𝑥𝐴𝑥)))
42, 3mpbiri 248 . 2 (𝐴𝑉𝐴 {𝑥 ∈ Tarski ∣ 𝐴𝑥})
5 tskmval 9774 . 2 (𝐴𝑉 → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
64, 5eleqtrrd 2806 1 (𝐴𝑉𝐴 ∈ (tarskiMap‘𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2103  ∀wral 3014  {crab 3018  ∩ cint 4583  ‘cfv 6001  Tarskictsk 9683  tarskiMapctskm 9772 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011  ax-groth 9758 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-int 4584  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-iota 5964  df-fun 6003  df-fv 6009  df-tsk 9684  df-tskm 9773 This theorem is referenced by:  eltskm  9778
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