Metamath Proof Explorer

Theorem trintALTVD

Description: The intersection of a class of transitive sets is transitive. Virtual deduction proof of trintALT . The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. trintALT is trintALTVD without virtual deductions and was automatically derived from trintALTVD .

1:: |- (. A. x e. A Tr x ->. A. x e. A Tr x ).
2:: |- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) ->. ( z e. y /\ y e. |^| A ) ).
3:2: |- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) ->. z e. y ).
4:2: |- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) ->. y e. |^| A ).
5:4: |- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) ->. A. q e. A y e. q ).
6:5: |- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) ->. ( q e. A -> y e. q ) ).
7:: |- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) , q e. A ->. q e. A ).
8:7,6: |- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) , q e. A ->. y e. q ).
9:7,1: |- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) , q e. A ->. [ q / x ] Tr x ).
10:7,9: |- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) , q e. A ->. Tr q ).
11:10,3,8: |- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) , q e. A ->. z e. q ).
12:11: |- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) ->. ( q e. A -> z e. q ) ).
13:12: |- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) ->. A. q ( q e. A -> z e. q ) ).
14:13: |- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) ->. A. q e. A z e. q ).
15:3,14: |- (. A. x e. A Tr x ,. ( z e. y /\ y e. |^| A ) ->. z e. |^| A ).
16:15: |- (. A. x e. A Tr x ->. ( ( z e. y /\ y e. |^| A ) -> z e. |^| A ) ).
17:16: |- (. A. x e. A Tr x ->. A. z A. y ( ( z e. y /\ y e. |^| A ) -> z e. |^| A ) ).
18:17: |- (. A. x e. A Tr x ->. Tr |^| A ).
qed:18: |- ( A. x e. A Tr x -> Tr |^| A )
(Contributed by Alan Sare, 17-Apr-2012) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion trintALTVD ( ∀ 𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴 )

Proof

Step Hyp Ref Expression
1 idn2 ⊢ (    ∀ 𝑥 ∈ 𝐴 Tr 𝑥    ,    ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴 )    ▶    ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴 )    )
2 simpl ⊢ ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴 ) → 𝑧 ∈ 𝑦 )
3 1 2 e2 ⊢ (    ∀ 𝑥 ∈ 𝐴 Tr 𝑥    ,    ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴 )    ▶    𝑧 ∈ 𝑦    )
4 idn3 ⊢ (    ∀ 𝑥 ∈ 𝐴 Tr 𝑥    ,    ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴 )    ,    𝑞 ∈ 𝐴    ▶    𝑞 ∈ 𝐴    )
5 idn1 ⊢ (    ∀ 𝑥 ∈ 𝐴 Tr 𝑥    ▶    ∀ 𝑥 ∈ 𝐴 Tr 𝑥    )
6 rspsbc ⊢ ( 𝑞 ∈ 𝐴 → ( ∀ 𝑥 ∈ 𝐴 Tr 𝑥 → [ 𝑞 / 𝑥 ] Tr 𝑥 ) )
7 4 5 6 e31 ⊢ (    ∀ 𝑥 ∈ 𝐴 Tr 𝑥    ,    ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴 )    ,    𝑞 ∈ 𝐴    ▶    [ 𝑞 / 𝑥 ] Tr 𝑥    )
8 trsbc ⊢ ( 𝑞 ∈ 𝐴 → ( [ 𝑞 / 𝑥 ] Tr 𝑥 ↔ Tr 𝑞 ) )
9 8 biimpd ⊢ ( 𝑞 ∈ 𝐴 → ( [ 𝑞 / 𝑥 ] Tr 𝑥 → Tr 𝑞 ) )
10 4 7 9 e33 ⊢ (    ∀ 𝑥 ∈ 𝐴 Tr 𝑥    ,    ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴 )    ,    𝑞 ∈ 𝐴    ▶    Tr 𝑞    )
11 simpr ⊢ ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴 ) → 𝑦 ∈ ∩ 𝐴 )
12 1 11 e2 ⊢ (    ∀ 𝑥 ∈ 𝐴 Tr 𝑥    ,    ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴 )    ▶    𝑦 ∈ ∩ 𝐴    )
13 elintg ⊢ ( 𝑦 ∈ ∩ 𝐴 → ( 𝑦 ∈ ∩ 𝐴 ↔ ∀ 𝑞 ∈ 𝐴 𝑦 ∈ 𝑞 ) )
14 13 ibi ⊢ ( 𝑦 ∈ ∩ 𝐴 → ∀ 𝑞 ∈ 𝐴 𝑦 ∈ 𝑞 )
15 12 14 e2 ⊢ (    ∀ 𝑥 ∈ 𝐴 Tr 𝑥    ,    ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴 )    ▶    ∀ 𝑞 ∈ 𝐴 𝑦 ∈ 𝑞    )
16 rsp ⊢ ( ∀ 𝑞 ∈ 𝐴 𝑦 ∈ 𝑞 → ( 𝑞 ∈ 𝐴 → 𝑦 ∈ 𝑞 ) )
17 15 16 e2 ⊢ (    ∀ 𝑥 ∈ 𝐴 Tr 𝑥    ,    ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴 )    ▶    ( 𝑞 ∈ 𝐴 → 𝑦 ∈ 𝑞 )    )
18 pm2.27 ⊢ ( 𝑞 ∈ 𝐴 → ( ( 𝑞 ∈ 𝐴 → 𝑦 ∈ 𝑞 ) → 𝑦 ∈ 𝑞 ) )
19 4 17 18 e32 ⊢ (    ∀ 𝑥 ∈ 𝐴 Tr 𝑥    ,    ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴 )    ,    𝑞 ∈ 𝐴    ▶    𝑦 ∈ 𝑞    )
20 trel ⊢ ( Tr 𝑞 → ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑞 ) → 𝑧 ∈ 𝑞 ) )
21 20 expd ⊢ ( Tr 𝑞 → ( 𝑧 ∈ 𝑦 → ( 𝑦 ∈ 𝑞 → 𝑧 ∈ 𝑞 ) ) )
22 10 3 19 21 e323 ⊢ (    ∀ 𝑥 ∈ 𝐴 Tr 𝑥    ,    ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴 )    ,    𝑞 ∈ 𝐴    ▶    𝑧 ∈ 𝑞    )
23 22 in3 ⊢ (    ∀ 𝑥 ∈ 𝐴 Tr 𝑥    ,    ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴 )    ▶    ( 𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞 )    )
24 23 gen21 ⊢ (    ∀ 𝑥 ∈ 𝐴 Tr 𝑥    ,    ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴 )    ▶    ∀ 𝑞 ( 𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞 )    )
25 df-ral ⊢ ( ∀ 𝑞 ∈ 𝐴 𝑧 ∈ 𝑞 ↔ ∀ 𝑞 ( 𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞 ) )
26 25 biimpri ⊢ ( ∀ 𝑞 ( 𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞 ) → ∀ 𝑞 ∈ 𝐴 𝑧 ∈ 𝑞 )
27 24 26 e2 ⊢ (    ∀ 𝑥 ∈ 𝐴 Tr 𝑥    ,    ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴 )    ▶    ∀ 𝑞 ∈ 𝐴 𝑧 ∈ 𝑞    )
28 elintg ⊢ ( 𝑧 ∈ 𝑦 → ( 𝑧 ∈ ∩ 𝐴 ↔ ∀ 𝑞 ∈ 𝐴 𝑧 ∈ 𝑞 ) )
29 28 biimprd ⊢ ( 𝑧 ∈ 𝑦 → ( ∀ 𝑞 ∈ 𝐴 𝑧 ∈ 𝑞 → 𝑧 ∈ ∩ 𝐴 ) )
30 3 27 29 e22 ⊢ (    ∀ 𝑥 ∈ 𝐴 Tr 𝑥    ,    ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴 )    ▶    𝑧 ∈ ∩ 𝐴    )
31 30 in2 ⊢ (    ∀ 𝑥 ∈ 𝐴 Tr 𝑥    ▶    ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴 ) → 𝑧 ∈ ∩ 𝐴 )    )
32 31 gen12 ⊢ (    ∀ 𝑥 ∈ 𝐴 Tr 𝑥    ▶    ∀ 𝑧 ∀ 𝑦 ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴 ) → 𝑧 ∈ ∩ 𝐴 )    )
33 dftr2 ⊢ ( Tr ∩ 𝐴 ↔ ∀ 𝑧 ∀ 𝑦 ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴 ) → 𝑧 ∈ ∩ 𝐴 ) )
34 33 biimpri ⊢ ( ∀ 𝑧 ∀ 𝑦 ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴 ) → 𝑧 ∈ ∩ 𝐴 ) → Tr ∩ 𝐴 )
35 32 34 e1a ⊢ (    ∀ 𝑥 ∈ 𝐴 Tr 𝑥    ▶    Tr ∩ 𝐴    )
36 35 in1 ⊢ ( ∀ 𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴 )