Theorem trintss 5230

Description: Any nonempty transitive class includes its intersection. Exercise 3 in [TakeutiZaring] p. 44 (which mistakenly does not include the nonemptiness hypothesis). (Contributed by Scott Fenton, 3-Mar-2011.) (Proof shortened by Andrew Salmon, 14-Nov-2011.)

Assertion

Ref	Expression
trintss	⊢ ((Tr 𝐴 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ 𝐴)

Proof of Theorem trintss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 4299	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2		intss1 4922	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥)
3		trss 5221	. . . . . 6 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
4	3	com12 33	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (Tr 𝐴 → 𝑥 ⊆ 𝐴))
5		sstr2 3915	. . . . 5 ⊢ (∩ 𝐴 ⊆ 𝑥 → (𝑥 ⊆ 𝐴 → ∩ 𝐴 ⊆ 𝐴))
6	2, 4, 5	sylsyld 62	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (Tr 𝐴 → ∩ 𝐴 ⊆ 𝐴))
7	6	exlimiv 1962	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (Tr 𝐴 → ∩ 𝐴 ⊆ 𝐴))
8	1, 7	sylbi 220	. 2 ⊢ (𝐴 ≠ ∅ → (Tr 𝐴 → ∩ 𝐴 ⊆ 𝐴))
9	8	impcom 413	1 ⊢ ((Tr 𝐴 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955   ⊆ wss 3895  ∅c0 4278  ∩ cint 4906  Tr wtr 5211

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-v 3452  df-ss 3896  df-dif 4047  df-nul 4279  df-uni 4867  df-int 4907  df-tr 5212

This theorem is used by: (None)