Theorem elnpi 10961

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 3478	. 2 ⊢ (𝐴 ∈ P → 𝐴 ∈ V)
2		simpl1 1208	. 2 ⊢ (((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)) → 𝐴 ∈ V)
3		psseq2 4047	. . . . . 6 ⊢ (𝑧 = 𝐴 → (∅ ⊊ 𝑧 ↔ ∅ ⊊ 𝐴))
4		psseq1 4046	. . . . . 6 ⊢ (𝑧 = 𝐴 → (𝑧 ⊊ Q ↔ 𝐴 ⊊ Q))
5	3, 4	anbi12d 643	. . . . 5 ⊢ (𝑧 = 𝐴 → ((∅ ⊊ 𝑧 ∧ 𝑧 ⊊ Q) ↔ (∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q)))
6		eleq2 2854	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝐴))
7	6	imbi2d 343	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → ((𝑦 <Q 𝑥 → 𝑦 ∈ 𝑧) ↔ (𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴)))
8	7	albidv 1943	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝑧) ↔ ∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴)))
9		rexeq 3319	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))
10	8, 9	anbi12d 643	. . . . . 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 3333	. . . . 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 643	. . . 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 10954	. . . 4 ⊢ P = {𝑧 ∣ ((∅ ⊊ 𝑧 ∧ 𝑧 ⊊ Q) ∧ ∀𝑥 ∈ 𝑧 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝑧) ∧ ∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦))}
14	12, 13	elab2g 3642	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ P ↔ ((∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))))
15		id 23	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) → (𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q))
16	15	3expib 1138	. . . . 5 ⊢ (𝐴 ∈ V → ((∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) → (𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q)))
17		3simpc 1166	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) → (∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q))
18	16, 17	impbid1 228	. . . 4 ⊢ (𝐴 ∈ V → ((∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ↔ (𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q)))
19	18	anbi1d 642	. . 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 282	. 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 381	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 400   ∧ w3a 1101  ∀wal 1561   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089  Vcvv 3457   ⊊ wpss 3908  ∅c0 4288   class class class wbr 5105  Qcnq 10825   <_Q cltq 10831  Pcnp 10832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-v 3459  df-ss 3924  df-pss 3927  df-np 10954

This theorem is referenced by:  prn0  10962  prpssnq  10963  prcdnq  10966  prnmax  10968