Theorem trint0m 3862

Description: Any inhabited transitive class includes its intersection. Similar to Exercise 2 in [TakeutiZaring] p. 44. (Contributed by Jim Kingdon, 22-Aug-2018.)

Assertion

Ref	Expression
trint0m	⊢ ((Tr A ∧ ∃x x ∈ A) → ∩ A ⊆ A)

Distinct variable group:   x,A

Proof of Theorem trint0m

Step	Hyp	Ref	Expression
1		intss1 3621	. . . 4 ⊢ (x ∈ A → ∩ A ⊆ x)
2		trss 3854	. . . . 5 ⊢ (Tr A → (x ∈ A → x ⊆ A))
3	2	com12 27	. . . 4 ⊢ (x ∈ A → (Tr A → x ⊆ A))
4		sstr2 2946	. . . 4 ⊢ (∩ A ⊆ x → (x ⊆ A → ∩ A ⊆ A))
5	1, 3, 4	sylsyld 52	. . 3 ⊢ (x ∈ A → (Tr A → ∩ A ⊆ A))
6	5	exlimiv 1486	. 2 ⊢ (∃x x ∈ A → (Tr A → ∩ A ⊆ A))
7	6	impcom 116	1 ⊢ ((Tr A ∧ ∃x x ∈ A) → ∩ A ⊆ A)

Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1378   ∈ wcel 1390   ⊆ wss 2911  ∩ cint 3606  Tr wtr 3845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-int 3607  df-tr 3846

This theorem is referenced by: (None)