Theorem trint0m 3868

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 3627	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥)
2		trss 3860	. . . . 5 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
3	2	com12 27	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (Tr 𝐴 → 𝑥 ⊆ 𝐴))
4		sstr2 2949	. . . 4 ⊢ (∩ 𝐴 ⊆ 𝑥 → (𝑥 ⊆ 𝐴 → ∩ 𝐴 ⊆ 𝐴))
5	1, 3, 4	sylsyld 52	. . 3 ⊢ (𝑥 ∈ 𝐴 → (Tr 𝐴 → ∩ 𝐴 ⊆ 𝐴))
6	5	exlimiv 1489	. 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 1381   ∈ wcel 1393   ⊆ wss 2914  ∩ cint 3612  Tr wtr 3851

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-v 2556  df-in 2921  df-ss 2928  df-uni 3578  df-int 3613  df-tr 3852

This theorem is referenced by: (None)