Metamath Proof Explorer

Theorem onfrALTlem1VD

Description: Virtual deduction proof of onfrALTlem1 . 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. onfrALTlem1 is onfrALTlem1VD without virtual deductions and was automatically derived from onfrALTlem1VD .

1:: |- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ ( a i^i x ) = (/) ) ->. ( x e. a /\ ( a i^i x ) = (/) ) ).
2:1: |- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ ( a i^i x ) = (/) ) ->. E. x ( x e. a /\ ( a i^i x ) = (/) ) ).
3:2: |- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ ( a i^i x ) = (/) ) ->. E. y [ y / x ] ( x e. a /\ ( a i^i x ) = (/) ) ).
4:: |- ( [ y / x ] ( x e. a /\ ( a i^i x ) = (/) ) <-> ( y e. a /\ ( a i^i y ) = (/) ) )
5:4: |- A. y ( [ y / x ] ( x e. a /\ ( a i^i x ) = (/) ) <-> ( y e. a /\ ( a i^i y ) = (/) ) )
6:5: |- ( E. y [ y / x ] ( x e. a /\ ( a i^i x ) = (/) ) <-> E. y ( y e. a /\ ( a i^i y ) = (/) ) )
7:3,6: |- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ ( a i^i x ) = (/) ) ->. E. y ( y e. a /\ ( a i^i y ) = (/) ) ).
8:: |- ( E. y e. a ( a i^i y ) = (/) <-> E. y ( y e. a /\ ( a i^i y ) = (/) ) )
qed:7,8: |- (. ( a C_ On /\ a =/= (/) ) ,. ( x e. a /\ ( a i^i x ) = (/) ) ->. E. y e. a ( a i^i y ) = (/) ).
(Contributed by Alan Sare, 22-Jul-2012) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion onfrALTlem1VD (    ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ )    ,    ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ )    ▶    ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅    )

Proof

Step Hyp Ref Expression
1 idn2 ⊢ (    ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ )    ,    ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ )    ▶    ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ )    )
2 19.8a ⊢ ( ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) → ∃ 𝑥 ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) )
3 1 2 e2 ⊢ (    ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ )    ,    ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ )    ▶    ∃ 𝑥 ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ )    )
4 cbvexsv ⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) ↔ ∃ 𝑦 [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) )
5 4 biimpi ⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) → ∃ 𝑦 [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) )
6 3 5 e2 ⊢ (    ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ )    ,    ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ )    ▶    ∃ 𝑦 [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ )    )
7 sbsbc ⊢ ( [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) ↔ [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) )
8 onfrALTlem4 ⊢ ( [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) ↔ ( 𝑦 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑦 ) = ∅ ) )
9 7 8 bitri ⊢ ( [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) ↔ ( 𝑦 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑦 ) = ∅ ) )
10 9 ax-gen ⊢ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) ↔ ( 𝑦 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑦 ) = ∅ ) )
11 exbi ⊢ ( ∀ 𝑦 ( [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) ↔ ( 𝑦 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑦 ) = ∅ ) ) → ( ∃ 𝑦 [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) ↔ ∃ 𝑦 ( 𝑦 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑦 ) = ∅ ) ) )
12 10 11 e0a ⊢ ( ∃ 𝑦 [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) ↔ ∃ 𝑦 ( 𝑦 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑦 ) = ∅ ) )
13 6 12 e2bi ⊢ (    ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ )    ,    ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ )    ▶    ∃ 𝑦 ( 𝑦 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑦 ) = ∅ )    )
14 df-rex ⊢ ( ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ ↔ ∃ 𝑦 ( 𝑦 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑦 ) = ∅ ) )
15 13 14 e2bir ⊢ (    ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ )    ,    ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ )    ▶    ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅    )