Theorem trintssOLD 4770

Description: Obsolete version of trintss 4769 as of 30-Oct-2021. (Contributed by Scott Fenton, 3-Mar-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem trintssOLD

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

Step	Hyp	Ref	Expression
1		vex 3203	. . . 4 ⊢ 𝑦 ∈ V
2	1	elint2 4482	. . 3 ⊢ (𝑦 ∈ ∩ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
3		r19.2z 4060	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
4	3	ex 450	. . . 4 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥))
5		trel 4759	. . . . . 6 ⊢ (Tr 𝐴 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
6	5	expcomd 454	. . . . 5 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
7	6	rexlimdv 3030	. . . 4 ⊢ (Tr 𝐴 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴))
8	4, 7	sylan9 689	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ Tr 𝐴) → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴))
9	2, 8	syl5bi 232	. 2 ⊢ ((𝐴 ≠ ∅ ∧ Tr 𝐴) → (𝑦 ∈ ∩ 𝐴 → 𝑦 ∈ 𝐴))
10	9	ssrdv 3609	1 ⊢ ((𝐴 ≠ ∅ ∧ Tr 𝐴) → ∩ 𝐴 ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574  ∅c0 3915  ∩ cint 4475  Tr wtr 4752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-uni 4437  df-int 4476  df-tr 4753

This theorem is referenced by: (None)