Theorem onfrALTVD 39127

Description: Virtual deduction proof of onfrALT 38764. 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 38764 is onfrALTVD 39127 without virtual deductions and was automatically derived from onfrALTVD 39127.

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 38790	. . . . . 6 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   )
2		simpr 477	. . . . . 6 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ≠ ∅)
3	1, 2	e1a 38852	. . . . 5 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑎 ≠ ∅   )
4		exmid 431	. . . . . . . . . 10 ⊢ ((𝑎 ∩ 𝑥) = ∅ ∨ ¬ (𝑎 ∩ 𝑥) = ∅)
5		onfrALTlem1VD 39126	. . . . . . . . . . 11 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅   )
6	5	in2an 38833	. . . . . . . . . 10 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥 ∈ 𝑎   ▶   ((𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)   )
7		onfrALTlem2VD 39125	. . . . . . . . . . 11 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅   )
8	7	in2an 38833	. . . . . . . . . 10 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥 ∈ 𝑎   ▶   (¬ (𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)   )
9		pm2.61 183	. . . . . . . . . . 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 38866	. . . . . . . . 9 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥 ∈ 𝑎   ▶   ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅   )
12	11	in2 38830	. . . . . . . 8 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)   )
13	12	gen11 38841	. . . . . . 7 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   ∀𝑥(𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)   )
14		19.23v 1902	. . . . . . . 8 ⊢ (∀𝑥(𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) ↔ (∃𝑥 𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))
15	14	biimpi 206	. . . . . . 7 ⊢ (∀𝑥(𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) → (∃𝑥 𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))
16	13, 15	e1a 38852	. . . . . 6 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (∃𝑥 𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)   )
17		n0 3931	. . . . . 6 ⊢ (𝑎 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑎)
18		imbi1 337	. . . . . . 7 ⊢ ((𝑎 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑎) → ((𝑎 ≠ ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) ↔ (∃𝑥 𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)))
19	18	biimprcd 240	. . . . . 6 ⊢ ((∃𝑥 𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) → ((𝑎 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑎) → (𝑎 ≠ ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)))
20	16, 17, 19	e10 38919	. . . . 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 38913	. . . 4 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅   )
23	22	in1 38787	. . 3 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)
24	23	ax-gen 1722	. 2 ⊢ ∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)
25		dfepfr 5099	. . 3 ⊢ ( E Fr On ↔ ∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))
26	25	biimpri 218	. 2 ⊢ (∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) → E Fr On)
27	24, 26	e0a 38999	1 ⊢ E Fr On

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915   E cep 5028   Fr wfr 5070  Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-vd1 38786  df-vd2 38794  df-vd3 38806

This theorem is referenced by: (None)