Theorem onfrALTlem1VD 44852

Description: Virtual deduction proof of onfrALTlem1 44511. 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 44511 is onfrALTlem1VD 44852 without virtual deductions and was automatically derived from onfrALTlem1VD 44852.

1::	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   )
2:1:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑥(𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   )
3:2:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦[𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)    )
4::	⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅ ) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
5:4:	⊢ ∀𝑦([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
6:5:	⊢ (∃𝑦[𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
7:3,6:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)   )
8::	⊢ (∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅ ↔ ∃𝑦( 𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
qed:7,8:	⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ 𝑎(𝑎 ∩ 𝑦) = ∅   )

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

Assertion

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

Distinct variable group:   𝑥,𝑎,𝑦

Proof of Theorem onfrALTlem1VD

Step	Hyp	Ref	Expression
1		idn2 44576	. . . . 5 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   )
2		19.8a 2182	. . . . 5 ⊢ ((𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) → ∃𝑥(𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅))
3	1, 2	e2 44594	. . . 4 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑥(𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   )
4		cbvexsv 44510	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ ∃𝑦[𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅))
5	4	biimpi 216	. . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦[𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅))
6	3, 5	e2 44594	. . 3 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦[𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   )
7		sbsbc 3754	. . . . . 6 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ [𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅))
8		onfrALTlem4 44506	. . . . . 6 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
9	7, 8	bitri 275	. . . . 5 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
10	9	ax-gen 1795	. . . 4 ⊢ ∀𝑦([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
11		exbi 1847	. . . 4 ⊢ (∀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 44734	. . 3 ⊢ (∃𝑦[𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
13	6, 12	e2bi 44595	. 2 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)   )
14		df-rex 3054	. 2 ⊢ (∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅ ↔ ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
15	13, 14	e2bir 44596	1 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅   )

Syntax hints:   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779  [wsb 2065   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053  [wsbc 3750   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  Oncon0 6320  (   wvd2 44540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-13 2370  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-in 3918  df-nul 4293  df-vd2 44541

This theorem is referenced by:  onfrALTVD  44853