Theorem trintss 4697

Description: If 𝐴 is transitive and non-null, then ∩ 𝐴 is a subset of 𝐴. (Contributed by Scott Fenton, 3-Mar-2011.)

Assertion

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

Proof of Theorem trintss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3176	. . . 4 ⊢ 𝑦 ∈ V
2	1	elint2 4417	. . 3 ⊢ (𝑦 ∈ ∩ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
3		r19.2z 4012	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
4	3	ex 449	. . . 4 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥))
5		trel 4687	. . . . . 6 ⊢ (Tr 𝐴 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
6	5	expcomd 453	. . . . 5 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
7	6	rexlimdv 3012	. . . 4 ⊢ (Tr 𝐴 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴))
8	4, 7	sylan9 687	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ Tr 𝐴) → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴))
9	2, 8	syl5bi 231	. 2 ⊢ ((𝐴 ≠ ∅ ∧ Tr 𝐴) → (𝑦 ∈ ∩ 𝐴 → 𝑦 ∈ 𝐴))
10	9	ssrdv 3574	1 ⊢ ((𝐴 ≠ ∅ ∧ Tr 𝐴) → ∩ 𝐴 ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ∅c0 3874  ∩ cint 4410  Tr wtr 4680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-uni 4373  df-int 4411  df-tr 4681

This theorem is referenced by: (None)