Theorem elnpi 10399

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 variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3459	. 2 ⊢ (𝐴 ∈ P → 𝐴 ∈ V)
2		simpl1 1188	. 2 ⊢ (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)) → 𝐴 ∈ V)
3		psseq2 4016	. . . . . 6 ⊢ (𝑧 = 𝑤 → (∅ ⊊ 𝑧 ↔ ∅ ⊊ 𝑤))
4		psseq1 4015	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑧 ⊊ Q ↔ 𝑤 ⊊ Q))
5	3, 4	anbi12d 633	. . . . 5 ⊢ (𝑧 = 𝑤 → ((∅ ⊊ 𝑧 ∧ 𝑧 ⊊ Q) ↔ (∅ ⊊ 𝑤 ∧ 𝑤 ⊊ Q)))
6		elequ2 2126	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑤))
7	6	imbi2d 344	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((𝑦 <Q 𝑥 → 𝑦 ∈ 𝑧) ↔ (𝑦 <Q 𝑥 → 𝑦 ∈ 𝑤)))
8	7	albidv 1921	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝑧) ↔ ∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝑤)))
9		rexeq 3359	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦 ↔ ∃𝑦 ∈ 𝑤 𝑥 <Q 𝑦))
10	8, 9	anbi12d 633	. . . . . 6 ⊢ (𝑧 = 𝑤 → ((∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝑧) ∧ ∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦) ↔ (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝑤) ∧ ∃𝑦 ∈ 𝑤 𝑥 <Q 𝑦)))
11	10	raleqbi1dv 3356	. . . . 5 ⊢ (𝑧 = 𝑤 → (∀𝑥 ∈ 𝑧 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝑧) ∧ ∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦) ↔ ∀𝑥 ∈ 𝑤 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝑤) ∧ ∃𝑦 ∈ 𝑤 𝑥 <Q 𝑦)))
12	5, 11	anbi12d 633	. . . 4 ⊢ (𝑧 = 𝑤 → (((∅ ⊊ 𝑧 ∧ 𝑧 ⊊ Q) ∧ ∀𝑥 ∈ 𝑧 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝑧) ∧ ∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦)) ↔ ((∅ ⊊ 𝑤 ∧ 𝑤 ⊊ Q) ∧ ∀𝑥 ∈ 𝑤 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝑤) ∧ ∃𝑦 ∈ 𝑤 𝑥 <Q 𝑦))))
13		psseq2 4016	. . . . . 6 ⊢ (𝑤 = 𝐴 → (∅ ⊊ 𝑤 ↔ ∅ ⊊ 𝐴))
14		psseq1 4015	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 ⊊ Q ↔ 𝐴 ⊊ Q))
15	13, 14	anbi12d 633	. . . . 5 ⊢ (𝑤 = 𝐴 → ((∅ ⊊ 𝑤 ∧ 𝑤 ⊊ Q) ↔ (∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q)))
16		eleq2 2878	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ 𝐴))
17	16	imbi2d 344	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → ((𝑦 <Q 𝑥 → 𝑦 ∈ 𝑤) ↔ (𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴)))
18	17	albidv 1921	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝑤) ↔ ∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴)))
19		rexeq 3359	. . . . . . 7 ⊢ (𝑤 = 𝐴 → (∃𝑦 ∈ 𝑤 𝑥 <Q 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))
20	18, 19	anbi12d 633	. . . . . 6 ⊢ (𝑤 = 𝐴 → ((∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝑤) ∧ ∃𝑦 ∈ 𝑤 𝑥 <Q 𝑦) ↔ (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)))
21	20	raleqbi1dv 3356	. . . . 5 ⊢ (𝑤 = 𝐴 → (∀𝑥 ∈ 𝑤 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝑤) ∧ ∃𝑦 ∈ 𝑤 𝑥 <Q 𝑦) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)))
22	15, 21	anbi12d 633	. . . 4 ⊢ (𝑤 = 𝐴 → (((∅ ⊊ 𝑤 ∧ 𝑤 ⊊ Q) ∧ ∀𝑥 ∈ 𝑤 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝑤) ∧ ∃𝑦 ∈ 𝑤 𝑥 <Q 𝑦)) ↔ ((∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))))
23		df-np 10392	. . . 4 ⊢ P = {𝑧 ∣ ((∅ ⊊ 𝑧 ∧ 𝑧 ⊊ Q) ∧ ∀𝑥 ∈ 𝑧 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝑧) ∧ ∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦))}
24	12, 22, 23	elab2gw 3613	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ P ↔ ((∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))))
25		id 22	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) → (𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q))
26	25	3expib 1119	. . . . 5 ⊢ (𝐴 ∈ V → ((∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) → (𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q)))
27		3simpc 1147	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) → (∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q))
28	26, 27	impbid1 228	. . . 4 ⊢ (𝐴 ∈ V → ((∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ↔ (𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q)))
29	28	anbi1d 632	. . 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))))
30	24, 29	bitrd 282	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))))
31	1, 2, 30	pm5.21nii 383	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 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ⊊ wpss 3882  ∅c0 4243   class class class wbr 5030  Qcnq 10263   <_Q cltq 10269  Pcnp 10270

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-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3443  df-in 3888  df-ss 3898  df-pss 3900  df-np 10392

This theorem is referenced by:  prn0  10400  prpssnq  10401  prcdnq  10404  prnmax  10406