Theorem onfrALTVD

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

	Ref	Expression
Assertion	onfrALTVD	⊢ E Fr On

Proof

Step	Hyp	Ref	Expression
1		idn1	⊢ (    ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ )    ▶    ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ )    )
2		simpr	⊢ ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → 𝑎 ≠ ∅ )
3	1 2	e1a	⊢ (    ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ )    ▶    𝑎 ≠ ∅    )
4		exmid	⊢ ( ( 𝑎 ∩ 𝑥 ) = ∅ ∨ ¬ ( 𝑎 ∩ 𝑥 ) = ∅ )
5		onfrALTlem1VD	⊢ (    ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ )    ,    ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ )    ▶    ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅    )
6	5	in2an	⊢ (    ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ )    ,    𝑥 ∈ 𝑎    ▶    ( ( 𝑎 ∩ 𝑥 ) = ∅ → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ )    )
7		onfrALTlem2VD	⊢ (    ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ )    ,    ( 𝑥 ∈ 𝑎 ∧ ¬ ( 𝑎 ∩ 𝑥 ) = ∅ )    ▶    ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅    )
8	7	in2an	⊢ (    ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ )    ,    𝑥 ∈ 𝑎    ▶    ( ¬ ( 𝑎 ∩ 𝑥 ) = ∅ → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ )    )
9		pm2.61	⊢ ( ( ( 𝑎 ∩ 𝑥 ) = ∅ → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ ) → ( ( ¬ ( 𝑎 ∩ 𝑥 ) = ∅ → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ ) → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ ) )
10	9	a1i	⊢ ( ( ( 𝑎 ∩ 𝑥 ) = ∅ ∨ ¬ ( 𝑎 ∩ 𝑥 ) = ∅ ) → ( ( ( 𝑎 ∩ 𝑥 ) = ∅ → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ ) → ( ( ¬ ( 𝑎 ∩ 𝑥 ) = ∅ → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ ) → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ ) ) )
11	4 6 8 10	e022	⊢ (    ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ )    ,    𝑥 ∈ 𝑎    ▶    ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅    )
12	11	in2	⊢ (    ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ )    ▶    ( 𝑥 ∈ 𝑎 → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ )    )
13	12	gen11	⊢ (    ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ )    ▶    ∀ 𝑥 ( 𝑥 ∈ 𝑎 → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ )    )
14		19.23v	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝑎 → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ ) ↔ ( ∃ 𝑥 𝑥 ∈ 𝑎 → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ ) )
15	14	biimpi	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝑎 → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ ) → ( ∃ 𝑥 𝑥 ∈ 𝑎 → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ ) )
16	13 15	e1a	⊢ (    ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ )    ▶    ( ∃ 𝑥 𝑥 ∈ 𝑎 → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ )    )
17		n0	⊢ ( 𝑎 ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝑎 )
18		imbi1	⊢ ( ( 𝑎 ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝑎 ) → ( ( 𝑎 ≠ ∅ → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ ) ↔ ( ∃ 𝑥 𝑥 ∈ 𝑎 → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ ) ) )
19	18	biimprcd	⊢ ( ( ∃ 𝑥 𝑥 ∈ 𝑎 → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ ) → ( ( 𝑎 ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝑎 ) → ( 𝑎 ≠ ∅ → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ ) ) )
20	16 17 19	e10	⊢ (    ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ )    ▶    ( 𝑎 ≠ ∅ → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ )    )
21		pm2.27	⊢ ( 𝑎 ≠ ∅ → ( ( 𝑎 ≠ ∅ → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ ) → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ ) )
22	3 20 21	e11	⊢ (    ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ )    ▶    ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅    )
23	22	in1	⊢ ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ )
24	23	ax-gen	⊢ ∀ 𝑎 ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ )
25		dfepfr	⊢ ( E Fr On ↔ ∀ 𝑎 ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ ) )
26	25	biimpri	⊢ ( ∀ 𝑎 ( ( 𝑎 ⊆ On ∧ 𝑎 ≠ ∅ ) → ∃ 𝑦 ∈ 𝑎 ( 𝑎 ∩ 𝑦 ) = ∅ ) → E Fr On )
27	24 26	e0a	⊢ E Fr On