Theorem prcdnq 10680

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 10613	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
2		relxp 5598	. . . . . . 7 ⊢ Rel (Q × Q)
3		relss 5682	. . . . . . 7 ⊢ ( <Q ⊆ (Q × Q) → (Rel (Q × Q) → Rel <Q ))
4	1, 2, 3	mp2 9	. . . . . 6 ⊢ Rel <Q
5	4	brrelex1i 5634	. . . . 5 ⊢ (𝐶 <Q 𝐵 → 𝐶 ∈ V)
6		eleq1 2826	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
7	6	anbi2d 628	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) ↔ (𝐴 ∈ P ∧ 𝐵 ∈ 𝐴)))
8		breq2 5074	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑦 <Q 𝑥 ↔ 𝑦 <Q 𝐵))
9	7, 8	anbi12d 630	. . . . . . 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 5073	. . . . . . . 8 ⊢ (𝑦 = 𝐶 → (𝑦 <Q 𝐵 ↔ 𝐶 <Q 𝐵))
12	11	anbi2d 628	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 <Q 𝐵) ↔ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 <Q 𝐵)))
13		eleq1 2826	. . . . . . 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 10675	. . . . . . . . . . 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 3132	. . . . . . . . 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 2184	. . . . . . 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 3500	. . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ V) → (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶 ∈ 𝐴))
22	5, 21	sylan2 592	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 <Q 𝐵) → (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶 ∈ 𝐴))
23	22	adantll 710	. . 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 1085  ∀wal 1537   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883   ⊊ wpss 3884  ∅c0 4253   class class class wbr 5070   × cxp 5578  Rel wrel 5585  Qcnq 10539   <_Q cltq 10545  Pcnp 10546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-ltnq 10605  df-np 10668

This theorem is referenced by:  prub  10681  addclprlem1  10703  mulclprlem  10706  distrlem4pr  10713  1idpr  10716  psslinpr  10718  prlem934  10720  ltaddpr  10721  ltexprlem2  10724  ltexprlem3  10725  ltexprlem6  10728  prlem936  10734  reclem2pr  10735  suplem1pr  10739