Theorem tpid3g 3831

Description: Closed theorem form of tpid3 3832. This proof was automatically generated from the virtual deduction proof tpid3gVD in set.mm using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)

Assertion

Ref	Expression
tpid3g	⊢ (A ∈ B → A ∈ {C, D, A})

Proof of Theorem tpid3g

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elisset 2869	. 2 ⊢ (A ∈ B → ∃x x = A)
2		3mix3 1126	. . . . . . 7 ⊢ (x = A → (x = C ∨ x = D ∨ x = A))
3	2	a1i 10	. . . . . 6 ⊢ (A ∈ B → (x = A → (x = C ∨ x = D ∨ x = A)))
4		abid 2341	. . . . . 6 ⊢ (x ∈ {x ∣ (x = C ∨ x = D ∨ x = A)} ↔ (x = C ∨ x = D ∨ x = A))
5	3, 4	syl6ibr 218	. . . . 5 ⊢ (A ∈ B → (x = A → x ∈ {x ∣ (x = C ∨ x = D ∨ x = A)}))
6		dftp2 3772	. . . . . 6 ⊢ {C, D, A} = {x ∣ (x = C ∨ x = D ∨ x = A)}
7	6	eleq2i 2417	. . . . 5 ⊢ (x ∈ {C, D, A} ↔ x ∈ {x ∣ (x = C ∨ x = D ∨ x = A)})
8	5, 7	syl6ibr 218	. . . 4 ⊢ (A ∈ B → (x = A → x ∈ {C, D, A}))
9		eleq1 2413	. . . 4 ⊢ (x = A → (x ∈ {C, D, A} ↔ A ∈ {C, D, A}))
10	8, 9	mpbidi 207	. . 3 ⊢ (A ∈ B → (x = A → A ∈ {C, D, A}))
11	10	exlimdv 1636	. 2 ⊢ (A ∈ B → (∃x x = A → A ∈ {C, D, A}))
12	1, 11	mpd 14	1 ⊢ (A ∈ B → A ∈ {C, D, A})

Syntax hints:   → wi 4   ∨ w3o 933  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  {ctp 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-tp 3743

This theorem is referenced by: (None)