Theorem tskmid 10596

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	⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ (tarskiMap‘𝐴))

Proof of Theorem tskmid

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)
2	1	rgenw 3076	. . 3 ⊢ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)
3		elintrabg 4892	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)))
4	2, 3	mpbiri 257	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
5		tskmval 10595	. 2 ⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
6	4, 5	eleqtrrd 2842	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ (tarskiMap‘𝐴))

Syntax hints:   → wi 4   ∈ wcel 2106  ∀wral 3064  {crab 3068  ∩ cint 4879  ‘cfv 6433  Tarskictsk 10504  tarskiMapctskm 10593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-groth 10579

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-tsk 10505  df-tskm 10594

This theorem is referenced by:  eltskm  10599