Theorem trintss 5238

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 4308	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2		intss1 4929	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑥)
3		trss 5229	. . . . . 6 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
4	3	com12 33	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (Tr 𝐴 → 𝑥 ⊆ 𝐴))
5		sstr2 3945	. . . . 5 ⊢ (∩ 𝐴 ⊆ 𝑥 → (𝑥 ⊆ 𝐴 → ∩ 𝐴 ⊆ 𝐴))
6	2, 4, 5	sylsyld 62	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (Tr 𝐴 → ∩ 𝐴 ⊆ 𝐴))
7	6	exlimiv 1960	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (Tr 𝐴 → ∩ 𝐴 ⊆ 𝐴))
8	1, 7	sylbi 220	. 2 ⊢ (𝐴 ≠ ∅ → (Tr 𝐴 → ∩ 𝐴 ⊆ 𝐴))
9	8	impcom 412	1 ⊢ ((Tr 𝐴 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 400  ∃wex 1809   ∈ wcel 2143   ≠ wne 2958   ⊆ wss 3906  ∅c0 4287  ∩ cint 4913  Tr wtr 5219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-v 3457  df-dif 3909  df-ss 3923  df-nul 4288  df-uni 4874  df-int 4914  df-tr 5220

This theorem is referenced by: (None)