Theorem prcdnq 10887

Description: A positive real is closed downwards under the positive fractions. Definition 9-3.1 (ii) of [Gleason] p. 121. (Contributed by NM, 25-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
prcdnq	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → (𝐶 <Q 𝐵 → 𝐶 ∈ 𝐴))

Proof of Theorem prcdnq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelnq 10820	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
2		relxp 5637	. . . . . . 7 ⊢ Rel (Q × Q)
3		relss 5725	. . . . . . 7 ⊢ ( <Q ⊆ (Q × Q) → (Rel (Q × Q) → Rel <Q ))
4	1, 2, 3	mp2 9	. . . . . 6 ⊢ Rel <Q
5	4	brrelex1i 5675	. . . . 5 ⊢ (𝐶 <Q 𝐵 → 𝐶 ∈ V)
6		eleq1 2816	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
7	6	anbi2d 630	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) ↔ (𝐴 ∈ P ∧ 𝐵 ∈ 𝐴)))
8		breq2 5096	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑦 <Q 𝑥 ↔ 𝑦 <Q 𝐵))
9	7, 8	anbi12d 632	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 <Q 𝑥) ↔ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 <Q 𝐵)))
10	9	imbi1d 341	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 <Q 𝑥) → 𝑦 ∈ 𝐴) ↔ (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 <Q 𝐵) → 𝑦 ∈ 𝐴)))
11		breq1 5095	. . . . . . . 8 ⊢ (𝑦 = 𝐶 → (𝑦 <Q 𝐵 ↔ 𝐶 <Q 𝐵))
12	11	anbi2d 630	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 <Q 𝐵) ↔ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 <Q 𝐵)))
13		eleq1 2816	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝑦 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
14	12, 13	imbi12d 344	. . . . . 6 ⊢ (𝑦 = 𝐶 → ((((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 <Q 𝐵) → 𝑦 ∈ 𝐴) ↔ (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶 ∈ 𝐴)))
15		elnpi 10882	. . . . . . . . . . 11 ⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)))
16	15	simprbi 496	. . . . . . . . . 10 ⊢ (𝐴 ∈ P → ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))
17	16	r19.21bi 3221	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) → (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))
18	17	simpld 494	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) → ∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴))
19	18	19.21bi 2190	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) → (𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴))
20	19	imp 406	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 <Q 𝑥) → 𝑦 ∈ 𝐴)
21	10, 14, 20	vtocl2g 3529	. . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ V) → (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶 ∈ 𝐴))
22	5, 21	sylan2 593	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 <Q 𝐵) → (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶 ∈ 𝐴))
23	22	adantll 714	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 <Q 𝐵) → (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶 ∈ 𝐴))
24	23	pm2.43i 52	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶 ∈ 𝐴)
25	24	ex 412	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → (𝐶 <Q 𝐵 → 𝐶 ∈ 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3436   ⊆ wss 3903   ⊊ wpss 3904  ∅c0 4284   class class class wbr 5092   × cxp 5617  Rel wrel 5624  Qcnq 10746   <_Q cltq 10752  Pcnp 10753

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-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

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-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-xp 5625  df-rel 5626  df-ltnq 10812  df-np 10875

This theorem is referenced by:  prub  10888  addclprlem1  10910  mulclprlem  10913  distrlem4pr  10920  1idpr  10923  psslinpr  10925  prlem934  10927  ltaddpr  10928  ltexprlem2  10931  ltexprlem3  10932  ltexprlem6  10935  prlem936  10941  reclem2pr  10942  suplem1pr  10946