Theorem trintssm 4117

Description: Any inhabited transitive class includes its intersection. Similar to Exercise 3 in [TakeutiZaring] p. 44 (which mistakenly does not include the inhabitedness hypothesis). (Contributed by Jim Kingdon, 22-Aug-2018.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem trintssm

Step	Hyp	Ref	Expression
1		intss1 3859	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥)
2		trss 4110	. . . . 5 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
3	2	com12 30	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (Tr 𝐴 → 𝑥 ⊆ 𝐴))
4		sstr2 3162	. . . 4 ⊢ (∩ 𝐴 ⊆ 𝑥 → (𝑥 ⊆ 𝐴 → ∩ 𝐴 ⊆ 𝐴))
5	1, 3, 4	sylsyld 58	. . 3 ⊢ (𝑥 ∈ 𝐴 → (Tr 𝐴 → ∩ 𝐴 ⊆ 𝐴))
6	5	exlimiv 1598	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (Tr 𝐴 → ∩ 𝐴 ⊆ 𝐴))
7	6	impcom 125	1 ⊢ ((Tr 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∩ 𝐴 ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 104  ∃wex 1492   ∈ wcel 2148   ⊆ wss 3129  ∩ cint 3844  Tr wtr 4101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-in 3135  df-ss 3142  df-uni 3810  df-int 3845  df-tr 4102

This theorem is referenced by: (None)