Theorem elnpi 10744

Description: Membership in positive reals. (Contributed by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
elnpi	⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)))

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem elnpi

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3450	. 2 ⊢ (𝐴 ∈ P → 𝐴 ∈ V)
2		simpl1 1190	. 2 ⊢ (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)) → 𝐴 ∈ V)
3		psseq2 4023	. . . . . 6 ⊢ (𝑧 = 𝐴 → (∅ ⊊ 𝑧 ↔ ∅ ⊊ 𝐴))
4		psseq1 4022	. . . . . 6 ⊢ (𝑧 = 𝐴 → (𝑧 ⊊ Q ↔ 𝐴 ⊊ Q))
5	3, 4	anbi12d 631	. . . . 5 ⊢ (𝑧 = 𝐴 → ((∅ ⊊ 𝑧 ∧ 𝑧 ⊊ Q) ↔ (∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q)))
6		eleq2 2827	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝐴))
7	6	imbi2d 341	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → ((𝑦 <Q 𝑥 → 𝑦 ∈ 𝑧) ↔ (𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴)))
8	7	albidv 1923	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝑧) ↔ ∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴)))
9		rexeq 3343	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))
10	8, 9	anbi12d 631	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝑧) ∧ ∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦) ↔ (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)))
11	10	raleqbi1dv 3340	. . . . 5 ⊢ (𝑧 = 𝐴 → (∀𝑥 ∈ 𝑧 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝑧) ∧ ∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)))
12	5, 11	anbi12d 631	. . . 4 ⊢ (𝑧 = 𝐴 → (((∅ ⊊ 𝑧 ∧ 𝑧 ⊊ Q) ∧ ∀𝑥 ∈ 𝑧 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝑧) ∧ ∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦)) ↔ ((∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))))
13		df-np 10737	. . . 4 ⊢ P = {𝑧 ∣ ((∅ ⊊ 𝑧 ∧ 𝑧 ⊊ Q) ∧ ∀𝑥 ∈ 𝑧 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝑧) ∧ ∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦))}
14	12, 13	elab2g 3611	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ P ↔ ((∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))))
15		id 22	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) → (𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q))
16	15	3expib 1121	. . . . 5 ⊢ (𝐴 ∈ V → ((∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) → (𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q)))
17		3simpc 1149	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) → (∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q))
18	16, 17	impbid1 224	. . . 4 ⊢ (𝐴 ∈ V → ((∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ↔ (𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q)))
19	18	anbi1d 630	. . 3 ⊢ (𝐴 ∈ V → (((∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)) ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))))
20	14, 19	bitrd 278	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))))
21	1, 2, 20	pm5.21nii 380	1 ⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086  ∀wal 1537   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ⊊ wpss 3888  ∅c0 4256   class class class wbr 5074  Qcnq 10608   <_Q cltq 10614  Pcnp 10615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-v 3434  df-in 3894  df-ss 3904  df-pss 3906  df-np 10737

This theorem is referenced by:  prn0  10745  prpssnq  10746  prcdnq  10749  prnmax  10751