Theorem onfrALTVD 45599

Description: Virtual deduction proof of onfrALT 45258. 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 45258 is onfrALTVD 45599 without virtual deductions and was automatically derived from onfrALTVD 45599.

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 45283	. . . . . 6 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   )
2		simpr 489	. . . . . 6 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → 𝑎 ≠ ∅)
3	1, 2	e1a 45336	. . . . 5 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   𝑎 ≠ ∅   )
4		exmid 907	. . . . . . . . . 10 ⊢ ((𝑎 ∩ 𝑥) = ∅ ∨ ¬ (𝑎 ∩ 𝑥) = ∅)
5		onfrALTlem1VD 45598	. . . . . . . . . . 11 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅   )
6	5	in2an 45317	. . . . . . . . . 10 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥 ∈ 𝑎   ▶   ((𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)   )
7		onfrALTlem2VD 45597	. . . . . . . . . . 11 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ ¬ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅   )
8	7	in2an 45317	. . . . . . . . . 10 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥 ∈ 𝑎   ▶   (¬ (𝑎 ∩ 𝑥) = ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)   )
9		pm2.61 194	. . . . . . . . . . 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 45350	. . . . . . . . 9 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   𝑥 ∈ 𝑎   ▶   ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅   )
12	11	in2 45314	. . . . . . . 8 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)   )
13	12	gen11 45325	. . . . . . 7 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   ∀𝑥(𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)   )
14		19.23v 1972	. . . . . . . 8 ⊢ (∀𝑥(𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) ↔ (∃𝑥 𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))
15	14	biimpi 219	. . . . . . 7 ⊢ (∀𝑥(𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) → (∃𝑥 𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))
16	13, 15	e1a 45336	. . . . . 6 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (∃𝑥 𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)   )
17		n0 4307	. . . . . 6 ⊢ (𝑎 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑎)
18		imbi1 350	. . . . . . 7 ⊢ ((𝑎 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑎) → ((𝑎 ≠ ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) ↔ (∃𝑥 𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)))
19	18	biimprcd 253	. . . . . 6 ⊢ ((∃𝑥 𝑥 ∈ 𝑎 → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) → ((𝑎 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑎) → (𝑎 ≠ ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)))
20	16, 17, 19	e10 45403	. . . . 5 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   (𝑎 ≠ ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)   )
21		pm2.27 43	. . . . 5 ⊢ (𝑎 ≠ ∅ → ((𝑎 ≠ ∅ → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))
22	3, 20, 21	e11 45397	. . . 4 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ▶   ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅   )
23	22	in1 45280	. . 3 ⊢ ((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)
24	23	ax-gen 1825	. 2 ⊢ ∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅)
25		dfepfr 5645	. . 3 ⊢ ( E Fr On ↔ ∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅))
26	25	biimpri 231	. 2 ⊢ (∀𝑎((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅) → E Fr On)
27	24, 26	e0a 45480	1 ⊢ E Fr On

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3089   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286   E cep 5560   Fr wfr 5611  Oncon0 6360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-13 2404  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-tr 5219  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-ord 6363  df-on 6364  df-vd1 45279  df-vd2 45287  df-vd3 45299

This theorem is referenced by: (None)