Theorem prcdnq 10407

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 10340	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
2		relxp 5566	. . . . . . 7 ⊢ Rel (Q × Q)
3		relss 5649	. . . . . . 7 ⊢ ( <Q ⊆ (Q × Q) → (Rel (Q × Q) → Rel <Q ))
4	1, 2, 3	mp2 9	. . . . . 6 ⊢ Rel <Q
5	4	brrelex1i 5601	. . . . 5 ⊢ (𝐶 <Q 𝐵 → 𝐶 ∈ V)
6		eleq1 2898	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
7	6	anbi2d 630	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) ↔ (𝐴 ∈ P ∧ 𝐵 ∈ 𝐴)))
8		breq2 5061	. . . . . . . 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 344	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 <Q 𝑥) → 𝑦 ∈ 𝐴) ↔ (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 <Q 𝐵) → 𝑦 ∈ 𝐴)))
11		breq1 5060	. . . . . . . 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 2898	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝑦 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
14	12, 13	imbi12d 347	. . . . . 6 ⊢ (𝑦 = 𝐶 → ((((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝑦 <Q 𝐵) → 𝑦 ∈ 𝐴) ↔ (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶 ∈ 𝐴)))
15		elnpi 10402	. . . . . . . . . . 11 ⊢ (𝐴 ∈ P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴 ∧ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦)))
16	15	simprbi 499	. . . . . . . . . 10 ⊢ (𝐴 ∈ P → ∀𝑥 ∈ 𝐴 (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))
17	16	r19.21bi 3206	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) → (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴) ∧ ∃𝑦 ∈ 𝐴 𝑥 <Q 𝑦))
18	17	simpld 497	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) → ∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴))
19	18	19.21bi 2180	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) → (𝑦 <Q 𝑥 → 𝑦 ∈ 𝐴))
20	19	imp 409	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 <Q 𝑥) → 𝑦 ∈ 𝐴)
21	10, 14, 20	vtocl2g 3570	. . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ V) → (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶 ∈ 𝐴))
22	5, 21	sylan2 594	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 <Q 𝐵) → (((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) ∧ 𝐶 <Q 𝐵) → 𝐶 ∈ 𝐴))
23	22	adantll 712	. . 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 415	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ 𝐴) → (𝐶 <Q 𝐵 → 𝐶 ∈ 𝐴))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1081  ∀wal 1528   = wceq 1530   ∈ wcel 2107  ∀wral 3136  ∃wrex 3137  Vcvv 3493   ⊆ wss 3934   ⊊ wpss 3935  ∅c0 4289   class class class wbr 5057   × cxp 5546  Rel wrel 5553  Qcnq 10266   <_Q cltq 10272  Pcnp 10273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-ltnq 10332  df-np 10395

This theorem is referenced by:  prub  10408  addclprlem1  10430  mulclprlem  10433  distrlem4pr  10440  1idpr  10443  psslinpr  10445  prlem934  10447  ltaddpr  10448  ltexprlem2  10451  ltexprlem3  10452  ltexprlem6  10455  prlem936  10461  reclem2pr  10462  suplem1pr  10466