Theorem trintssm 4208

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 3948	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥)
2		trss 4201	. . . . 5 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
3	2	com12 30	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (Tr 𝐴 → 𝑥 ⊆ 𝐴))
4		sstr2 3235	. . . 4 ⊢ (∩ 𝐴 ⊆ 𝑥 → (𝑥 ⊆ 𝐴 → ∩ 𝐴 ⊆ 𝐴))
5	1, 3, 4	sylsyld 58	. . 3 ⊢ (𝑥 ∈ 𝐴 → (Tr 𝐴 → ∩ 𝐴 ⊆ 𝐴))
6	5	exlimiv 1647	. 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 1541   ∈ wcel 2202   ⊆ wss 3201  ∩ cint 3933  Tr wtr 4192

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-in 3207  df-ss 3214  df-uni 3899  df-int 3934  df-tr 4193

This theorem is referenced by: (None)