Theorem onfrALTVD 44916

Description: Virtual deduction proof of onfrALT 44574. 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 44574 is onfrALTVD 44916 without virtual deductions and was automatically derived from onfrALTVD 44916.

1::	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅   )
2::	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅   )
3:1:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥 ∈ 𝑎   ▶    (¬ (𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅)   )
4:2:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥 ∈ 𝑎   ▶    ((𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅)   )
5::	⊢ ((𝑎 ∩ 𝑥) = ∅ ∨ ¬ (𝑎 ∩ 𝑥) = ∅)
6:5,4,3:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥 ∈ 𝑎   ▶    ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅   )
7:6:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅)   )
8:7:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   ∀𝑥(𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅)   )
9:8:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (∃𝑥𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅)   )
10::	⊢ (𝑎 ≠ ∅ ↔ ∃𝑥𝑥 ∈ 𝑎)
11:9,10:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ≠ ∅ → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅)   )
12::	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   )
13:12:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑎 ≠ ∅   )
14:13,11:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅   )
15:14:	⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)
16:15:	⊢ ∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅)
qed:16:	⊢ E Fr On

(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
onfrALTVD	⊢ E Fr On

Proof of Theorem onfrALTVD

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

Step	Hyp	Ref	Expression
1		idn1 44599	. . . . . 6 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   )
2		simpr 484	. . . . . 6 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ≠ ∅)
3	1, 2	e1a 44652	. . . . 5 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑎 ≠ ∅   )
4		exmid 894	. . . . . . . . . 10 ⊢ ((𝑎 ∩ 𝑥) = ∅ ∨ ¬ (𝑎 ∩ 𝑥) = ∅)
5		onfrALTlem1VD 44915	. . . . . . . . . . 11 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅   )
6	5	in2an 44633	. . . . . . . . . 10 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥 ∈ 𝑎   ▶   ((𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)   )
7		onfrALTlem2VD 44914	. . . . . . . . . . 11 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅   )
8	7	in2an 44633	. . . . . . . . . 10 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥 ∈ 𝑎   ▶   (¬ (𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)   )
9		pm2.61 192	. . . . . . . . . . 11 ⊢ (((𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) → ((¬ (𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))
10	9	a1i 11	. . . . . . . . . 10 ⊢ (((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 44666	. . . . . . . . 9 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥 ∈ 𝑎   ▶   ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅   )
12	11	in2 44630	. . . . . . . 8 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)   )
13	12	gen11 44641	. . . . . . 7 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   ∀𝑥(𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)   )
14		19.23v 1941	. . . . . . . 8 ⊢ (∀𝑥(𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) ↔ (∃𝑥 𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))
15	14	biimpi 216	. . . . . . 7 ⊢ (∀𝑥(𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) → (∃𝑥 𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))
16	13, 15	e1a 44652	. . . . . 6 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (∃𝑥 𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)   )
17		n0 4352	. . . . . 6 ⊢ (𝑎 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑎)
18		imbi1 347	. . . . . . 7 ⊢ ((𝑎 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑎) → ((𝑎 ≠ ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) ↔ (∃𝑥 𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)))
19	18	biimprcd 250	. . . . . 6 ⊢ ((∃𝑥 𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) → ((𝑎 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑎) → (𝑎 ≠ ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)))
20	16, 17, 19	e10 44719	. . . . 5 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ≠ ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)   )
21		pm2.27 42	. . . . 5 ⊢ (𝑎 ≠ ∅ → ((𝑎 ≠ ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))
22	3, 20, 21	e11 44713	. . . 4 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅   )
23	22	in1 44596	. . 3 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)
24	23	ax-gen 1794	. 2 ⊢ ∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)
25		dfepfr 5668	. . 3 ⊢ ( E Fr On ↔ ∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))
26	25	biimpri 228	. 2 ⊢ (∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) → E Fr On)
27	24, 26	e0a 44797	1 ⊢ E Fr On

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847  ∀wal 1537   = wceq 1539  ∃wex 1778   ∈ wcel 2107   ≠ wne 2939  ∃wrex 3069   ∩ cin 3949   ⊆ wss 3950  ∅c0 4332   E cep 5582   Fr wfr 5633  Oncon0 6383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-13 2376  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387  df-vd1 44595  df-vd2 44603  df-vd3 44615

This theorem is referenced by: (None)