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 ( ∀ 𝑥𝐴 Tr 𝑥 → Tr 𝐴 )

Proof

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