Theorem onfrALTlem1VD 41596

Description: Virtual deduction proof of onfrALTlem1 41254. 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 41254 is onfrALTlem1VD 41596 without virtual deductions and was automatically derived from onfrALTlem1VD 41596.

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 41319	. . . . 5 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   )
2		19.8a 2178	. . . . 5 ⊢ ((𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) → ∃𝑥(𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅))
3	1, 2	e2 41337	. . . 4 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑥(𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   )
4		cbvexsv 41253	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ ∃𝑦[𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅))
5	4	biimpi 219	. . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) → ∃𝑦[𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅))
6	3, 5	e2 41337	. . 3 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦[𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   )
7		sbsbc 3724	. . . . . 6 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ [𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅))
8		onfrALTlem4 41249	. . . . . 6 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
9	7, 8	bitri 278	. . . . 5 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
10	9	ax-gen 1797	. . . 4 ⊢ ∀𝑦([𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ (𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
11		exbi 1848	. . . 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 41478	. . 3 ⊢ (∃𝑦[𝑦 / 𝑥](𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅) ↔ ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
13	6, 12	e2bi 41338	. 2 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅)   )
14		df-rex 3112	. 2 ⊢ (∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅ ↔ ∃𝑦(𝑦 ∈ 𝑎 ∧ (𝑎 ∩ 𝑦) = ∅))
15	13, 14	e2bir 41339	1 ⊢ (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥 ∈ 𝑎 ∧ (𝑎 ∩ 𝑥) = ∅)   ▶   ∃𝑦 ∈ 𝑎 (𝑎 ∩ 𝑦) = ∅   )

Syntax hints:   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781  [wsb 2069   ∈ wcel 2111   ≠ wne 2987  ∃wrex 3107  [wsbc 3720   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  Oncon0 6159  (   wvd2 41283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-in 3888  df-nul 4244  df-vd2 41284

This theorem is referenced by:  onfrALTVD  41597