Theorem suplem1pr 9753

Description: The union of a nonempty, bounded set of positive reals is a positive real. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
suplem1pr	⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥) → ∪ 𝐴 ∈ P)

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem suplem1pr

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelpr 9699	. . . . . . . . 9 ⊢ <P ⊆ (P × P)
2	1	brel 5090	. . . . . . . 8 ⊢ (𝑦<P 𝑥 → (𝑦 ∈ P ∧ 𝑥 ∈ P))
3	2	simpld 474	. . . . . . 7 ⊢ (𝑦<P 𝑥 → 𝑦 ∈ P)
4	3	ralimi 2936	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 𝑦<P 𝑥 → ∀𝑦 ∈ 𝐴 𝑦 ∈ P)
5		dfss3 3558	. . . . . 6 ⊢ (𝐴 ⊆ P ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ P)
6	4, 5	sylibr 223	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 𝑦<P 𝑥 → 𝐴 ⊆ P)
7	6	rexlimivw 3011	. . . 4 ⊢ (∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥 → 𝐴 ⊆ P)
8	7	adantl 481	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥) → 𝐴 ⊆ P)
9		n0 3890	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐴)
10		ssel 3562	. . . . . . 7 ⊢ (𝐴 ⊆ P → (𝑧 ∈ 𝐴 → 𝑧 ∈ P))
11		prn0 9690	. . . . . . . . . 10 ⊢ (𝑧 ∈ P → 𝑧 ≠ ∅)
12		0pss 3965	. . . . . . . . . 10 ⊢ (∅ ⊊ 𝑧 ↔ 𝑧 ≠ ∅)
13	11, 12	sylibr 223	. . . . . . . . 9 ⊢ (𝑧 ∈ P → ∅ ⊊ 𝑧)
14		elssuni 4403	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ⊆ ∪ 𝐴)
15		psssstr 3675	. . . . . . . . 9 ⊢ ((∅ ⊊ 𝑧 ∧ 𝑧 ⊆ ∪ 𝐴) → ∅ ⊊ ∪ 𝐴)
16	13, 14, 15	syl2an 493	. . . . . . . 8 ⊢ ((𝑧 ∈ P ∧ 𝑧 ∈ 𝐴) → ∅ ⊊ ∪ 𝐴)
17	16	expcom 450	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → (𝑧 ∈ P → ∅ ⊊ ∪ 𝐴))
18	10, 17	sylcom 30	. . . . . 6 ⊢ (𝐴 ⊆ P → (𝑧 ∈ 𝐴 → ∅ ⊊ ∪ 𝐴))
19	18	exlimdv 1848	. . . . 5 ⊢ (𝐴 ⊆ P → (∃𝑧 𝑧 ∈ 𝐴 → ∅ ⊊ ∪ 𝐴))
20	9, 19	syl5bi 231	. . . 4 ⊢ (𝐴 ⊆ P → (𝐴 ≠ ∅ → ∅ ⊊ ∪ 𝐴))
21		prpssnq 9691	. . . . . . 7 ⊢ (𝑥 ∈ P → 𝑥 ⊊ Q)
22	21	adantl 481	. . . . . 6 ⊢ ((𝐴 ⊆ P ∧ 𝑥 ∈ P) → 𝑥 ⊊ Q)
23		ltprord 9731	. . . . . . . . . 10 ⊢ ((𝑦 ∈ P ∧ 𝑥 ∈ P) → (𝑦<P 𝑥 ↔ 𝑦 ⊊ 𝑥))
24		pssss 3664	. . . . . . . . . 10 ⊢ (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝑥)
25	23, 24	syl6bi 242	. . . . . . . . 9 ⊢ ((𝑦 ∈ P ∧ 𝑥 ∈ P) → (𝑦<P 𝑥 → 𝑦 ⊆ 𝑥))
26	2, 25	mpcom 37	. . . . . . . 8 ⊢ (𝑦<P 𝑥 → 𝑦 ⊆ 𝑥)
27	26	ralimi 2936	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐴 𝑦<P 𝑥 → ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)
28		unissb 4405	. . . . . . 7 ⊢ (∪ 𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)
29	27, 28	sylibr 223	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 𝑦<P 𝑥 → ∪ 𝐴 ⊆ 𝑥)
30		sspsstr 3674	. . . . . . 7 ⊢ ((∪ 𝐴 ⊆ 𝑥 ∧ 𝑥 ⊊ Q) → ∪ 𝐴 ⊊ Q)
31	30	expcom 450	. . . . . 6 ⊢ (𝑥 ⊊ Q → (∪ 𝐴 ⊆ 𝑥 → ∪ 𝐴 ⊊ Q))
32	22, 29, 31	syl2im 39	. . . . 5 ⊢ ((𝐴 ⊆ P ∧ 𝑥 ∈ P) → (∀𝑦 ∈ 𝐴 𝑦<P 𝑥 → ∪ 𝐴 ⊊ Q))
33	32	rexlimdva 3013	. . . 4 ⊢ (𝐴 ⊆ P → (∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥 → ∪ 𝐴 ⊊ Q))
34	20, 33	anim12d 584	. . 3 ⊢ (𝐴 ⊆ P → ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥) → (∅ ⊊ ∪ 𝐴 ∧ ∪ 𝐴 ⊊ Q)))
35	8, 34	mpcom 37	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥) → (∅ ⊊ ∪ 𝐴 ∧ ∪ 𝐴 ⊊ Q))
36		prcdnq 9694	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ P ∧ 𝑥 ∈ 𝑧) → (𝑦 <Q 𝑥 → 𝑦 ∈ 𝑧))
37	36	ex 449	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ P → (𝑥 ∈ 𝑧 → (𝑦 <Q 𝑥 → 𝑦 ∈ 𝑧)))
38	37	com3r 85	. . . . . . . . . . 11 ⊢ (𝑦 <Q 𝑥 → (𝑧 ∈ P → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)))
39	10, 38	sylan9 687	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ P ∧ 𝑦 <Q 𝑥) → (𝑧 ∈ 𝐴 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)))
40	39	reximdvai 2998	. . . . . . . . 9 ⊢ ((𝐴 ⊆ P ∧ 𝑦 <Q 𝑥) → (∃𝑧 ∈ 𝐴 𝑥 ∈ 𝑧 → ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧))
41		eluni2 4376	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑧 ∈ 𝐴 𝑥 ∈ 𝑧)
42		eluni2 4376	. . . . . . . . 9 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧)
43	40, 41, 42	3imtr4g 284	. . . . . . . 8 ⊢ ((𝐴 ⊆ P ∧ 𝑦 <Q 𝑥) → (𝑥 ∈ ∪ 𝐴 → 𝑦 ∈ ∪ 𝐴))
44	43	ex 449	. . . . . . 7 ⊢ (𝐴 ⊆ P → (𝑦 <Q 𝑥 → (𝑥 ∈ ∪ 𝐴 → 𝑦 ∈ ∪ 𝐴)))
45	44	com23 84	. . . . . 6 ⊢ (𝐴 ⊆ P → (𝑥 ∈ ∪ 𝐴 → (𝑦 <Q 𝑥 → 𝑦 ∈ ∪ 𝐴)))
46	45	alrimdv 1844	. . . . 5 ⊢ (𝐴 ⊆ P → (𝑥 ∈ ∪ 𝐴 → ∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ ∪ 𝐴)))
47		eluni 4375	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴))
48		prnmax 9696	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ P ∧ 𝑥 ∈ 𝑧) → ∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦)
49	48	ex 449	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ P → (𝑥 ∈ 𝑧 → ∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦))
50	10, 49	syl6 34	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ P → (𝑧 ∈ 𝐴 → (𝑥 ∈ 𝑧 → ∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦)))
51	50	com23 84	. . . . . . . . . 10 ⊢ (𝐴 ⊆ P → (𝑥 ∈ 𝑧 → (𝑧 ∈ 𝐴 → ∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦)))
52	51	imp 444	. . . . . . . . 9 ⊢ ((𝐴 ⊆ P ∧ 𝑥 ∈ 𝑧) → (𝑧 ∈ 𝐴 → ∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦))
53		ssrexv 3630	. . . . . . . . . 10 ⊢ (𝑧 ⊆ ∪ 𝐴 → (∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦 → ∃𝑦 ∈ ∪ 𝐴𝑥 <Q 𝑦))
54	14, 53	syl 17	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 → (∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦 → ∃𝑦 ∈ ∪ 𝐴𝑥 <Q 𝑦))
55	52, 54	sylcom 30	. . . . . . . 8 ⊢ ((𝐴 ⊆ P ∧ 𝑥 ∈ 𝑧) → (𝑧 ∈ 𝐴 → ∃𝑦 ∈ ∪ 𝐴𝑥 <Q 𝑦))
56	55	expimpd 627	. . . . . . 7 ⊢ (𝐴 ⊆ P → ((𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → ∃𝑦 ∈ ∪ 𝐴𝑥 <Q 𝑦))
57	56	exlimdv 1848	. . . . . 6 ⊢ (𝐴 ⊆ P → (∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → ∃𝑦 ∈ ∪ 𝐴𝑥 <Q 𝑦))
58	47, 57	syl5bi 231	. . . . 5 ⊢ (𝐴 ⊆ P → (𝑥 ∈ ∪ 𝐴 → ∃𝑦 ∈ ∪ 𝐴𝑥 <Q 𝑦))
59	46, 58	jcad 554	. . . 4 ⊢ (𝐴 ⊆ P → (𝑥 ∈ ∪ 𝐴 → (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ ∪ 𝐴) ∧ ∃𝑦 ∈ ∪ 𝐴𝑥 <Q 𝑦)))
60	59	ralrimiv 2948	. . 3 ⊢ (𝐴 ⊆ P → ∀𝑥 ∈ ∪ 𝐴(∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ ∪ 𝐴) ∧ ∃𝑦 ∈ ∪ 𝐴𝑥 <Q 𝑦))
61	8, 60	syl 17	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥) → ∀𝑥 ∈ ∪ 𝐴(∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ ∪ 𝐴) ∧ ∃𝑦 ∈ ∪ 𝐴𝑥 <Q 𝑦))
62		elnp 9688	. 2 ⊢ (∪ 𝐴 ∈ P ↔ ((∅ ⊊ ∪ 𝐴 ∧ ∪ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ ∪ 𝐴(∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ ∪ 𝐴) ∧ ∃𝑦 ∈ ∪ 𝐴𝑥 <Q 𝑦)))
63	35, 61, 62	sylanbrc 695	1 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥) → ∪ 𝐴 ∈ P)

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540   ⊊ wpss 3541  ∅c0 3874  ∪ cuni 4372   class class class wbr 4583  Qcnq 9553   <_Q cltq 9559  Pcnp 9560  <_P cltp 9564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-om 6958  df-ni 9573  df-nq 9613  df-ltnq 9619  df-np 9682  df-ltp 9686

This theorem is referenced by:  supexpr  9755