Theorem suplem1pr 11090

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 11036	. . . . . . . . 9 ⊢ <P ⊆ (P × P)
2	1	brel 5754	. . . . . . . 8 ⊢ (𝑦<P 𝑥 → (𝑦 ∈ P ∧ 𝑥 ∈ P))
3	2	simpld 494	. . . . . . 7 ⊢ (𝑦<P 𝑥 → 𝑦 ∈ P)
4	3	ralimi 3081	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 𝑦<P 𝑥 → ∀𝑦 ∈ 𝐴 𝑦 ∈ P)
5		dfss3 3984	. . . . . 6 ⊢ (𝐴 ⊆ P ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ P)
6	4, 5	sylibr 234	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 𝑦<P 𝑥 → 𝐴 ⊆ P)
7	6	rexlimivw 3149	. . . 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 4359	. . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐴)
10		ssel 3989	. . . . . . 7 ⊢ (𝐴 ⊆ P → (𝑧 ∈ 𝐴 → 𝑧 ∈ P))
11		prn0 11027	. . . . . . . . . 10 ⊢ (𝑧 ∈ P → 𝑧 ≠ ∅)
12		0pss 4453	. . . . . . . . . 10 ⊢ (∅ ⊊ 𝑧 ↔ 𝑧 ≠ ∅)
13	11, 12	sylibr 234	. . . . . . . . 9 ⊢ (𝑧 ∈ P → ∅ ⊊ 𝑧)
14		elssuni 4942	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ⊆ ∪ 𝐴)
15		psssstr 4119	. . . . . . . . 9 ⊢ ((∅ ⊊ 𝑧 ∧ 𝑧 ⊆ ∪ 𝐴) → ∅ ⊊ ∪ 𝐴)
16	13, 14, 15	syl2an 596	. . . . . . . 8 ⊢ ((𝑧 ∈ P ∧ 𝑧 ∈ 𝐴) → ∅ ⊊ ∪ 𝐴)
17	16	expcom 413	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → (𝑧 ∈ P → ∅ ⊊ ∪ 𝐴))
18	10, 17	sylcom 30	. . . . . 6 ⊢ (𝐴 ⊆ P → (𝑧 ∈ 𝐴 → ∅ ⊊ ∪ 𝐴))
19	18	exlimdv 1931	. . . . 5 ⊢ (𝐴 ⊆ P → (∃𝑧 𝑧 ∈ 𝐴 → ∅ ⊊ ∪ 𝐴))
20	9, 19	biimtrid 242	. . . 4 ⊢ (𝐴 ⊆ P → (𝐴 ≠ ∅ → ∅ ⊊ ∪ 𝐴))
21		prpssnq 11028	. . . . . . 7 ⊢ (𝑥 ∈ P → 𝑥 ⊊ Q)
22	21	adantl 481	. . . . . 6 ⊢ ((𝐴 ⊆ P ∧ 𝑥 ∈ P) → 𝑥 ⊊ Q)
23		ltprord 11068	. . . . . . . . . 10 ⊢ ((𝑦 ∈ P ∧ 𝑥 ∈ P) → (𝑦<P 𝑥 ↔ 𝑦 ⊊ 𝑥))
24		pssss 4108	. . . . . . . . . 10 ⊢ (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝑥)
25	23, 24	biimtrdi 253	. . . . . . . . 9 ⊢ ((𝑦 ∈ P ∧ 𝑥 ∈ P) → (𝑦<P 𝑥 → 𝑦 ⊆ 𝑥))
26	2, 25	mpcom 38	. . . . . . . 8 ⊢ (𝑦<P 𝑥 → 𝑦 ⊆ 𝑥)
27	26	ralimi 3081	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐴 𝑦<P 𝑥 → ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)
28		unissb 4944	. . . . . . 7 ⊢ (∪ 𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)
29	27, 28	sylibr 234	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 𝑦<P 𝑥 → ∪ 𝐴 ⊆ 𝑥)
30		sspsstr 4118	. . . . . . 7 ⊢ ((∪ 𝐴 ⊆ 𝑥 ∧ 𝑥 ⊊ Q) → ∪ 𝐴 ⊊ Q)
31	30	expcom 413	. . . . . 6 ⊢ (𝑥 ⊊ Q → (∪ 𝐴 ⊆ 𝑥 → ∪ 𝐴 ⊊ Q))
32	22, 29, 31	syl2im 40	. . . . 5 ⊢ ((𝐴 ⊆ P ∧ 𝑥 ∈ P) → (∀𝑦 ∈ 𝐴 𝑦<P 𝑥 → ∪ 𝐴 ⊊ Q))
33	32	rexlimdva 3153	. . . 4 ⊢ (𝐴 ⊆ P → (∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥 → ∪ 𝐴 ⊊ Q))
34	20, 33	anim12d 609	. . 3 ⊢ (𝐴 ⊆ P → ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥) → (∅ ⊊ ∪ 𝐴 ∧ ∪ 𝐴 ⊊ Q)))
35	8, 34	mpcom 38	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥) → (∅ ⊊ ∪ 𝐴 ∧ ∪ 𝐴 ⊊ Q))
36		prcdnq 11031	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ P ∧ 𝑥 ∈ 𝑧) → (𝑦 <Q 𝑥 → 𝑦 ∈ 𝑧))
37	36	ex 412	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ P → (𝑥 ∈ 𝑧 → (𝑦 <Q 𝑥 → 𝑦 ∈ 𝑧)))
38	37	com3r 87	. . . . . . . . . . 11 ⊢ (𝑦 <Q 𝑥 → (𝑧 ∈ P → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)))
39	10, 38	sylan9 507	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ P ∧ 𝑦 <Q 𝑥) → (𝑧 ∈ 𝐴 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)))
40	39	reximdvai 3163	. . . . . . . . 9 ⊢ ((𝐴 ⊆ P ∧ 𝑦 <Q 𝑥) → (∃𝑧 ∈ 𝐴 𝑥 ∈ 𝑧 → ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧))
41		eluni2 4916	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑧 ∈ 𝐴 𝑥 ∈ 𝑧)
42		eluni2 4916	. . . . . . . . 9 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑧 ∈ 𝐴 𝑦 ∈ 𝑧)
43	40, 41, 42	3imtr4g 296	. . . . . . . 8 ⊢ ((𝐴 ⊆ P ∧ 𝑦 <Q 𝑥) → (𝑥 ∈ ∪ 𝐴 → 𝑦 ∈ ∪ 𝐴))
44	43	ex 412	. . . . . . 7 ⊢ (𝐴 ⊆ P → (𝑦 <Q 𝑥 → (𝑥 ∈ ∪ 𝐴 → 𝑦 ∈ ∪ 𝐴)))
45	44	com23 86	. . . . . 6 ⊢ (𝐴 ⊆ P → (𝑥 ∈ ∪ 𝐴 → (𝑦 <Q 𝑥 → 𝑦 ∈ ∪ 𝐴)))
46	45	alrimdv 1927	. . . . 5 ⊢ (𝐴 ⊆ P → (𝑥 ∈ ∪ 𝐴 → ∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ ∪ 𝐴)))
47		eluni 4915	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴))
48		prnmax 11033	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ P ∧ 𝑥 ∈ 𝑧) → ∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦)
49	48	ex 412	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ P → (𝑥 ∈ 𝑧 → ∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦))
50	10, 49	syl6 35	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ P → (𝑧 ∈ 𝐴 → (𝑥 ∈ 𝑧 → ∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦)))
51	50	com23 86	. . . . . . . . . 10 ⊢ (𝐴 ⊆ P → (𝑥 ∈ 𝑧 → (𝑧 ∈ 𝐴 → ∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦)))
52	51	imp 406	. . . . . . . . 9 ⊢ ((𝐴 ⊆ P ∧ 𝑥 ∈ 𝑧) → (𝑧 ∈ 𝐴 → ∃𝑦 ∈ 𝑧 𝑥 <Q 𝑦))
53		ssrexv 4065	. . . . . . . . . 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 453	. . . . . . 7 ⊢ (𝐴 ⊆ P → ((𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → ∃𝑦 ∈ ∪ 𝐴𝑥 <Q 𝑦))
57	56	exlimdv 1931	. . . . . 6 ⊢ (𝐴 ⊆ P → (∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → ∃𝑦 ∈ ∪ 𝐴𝑥 <Q 𝑦))
58	47, 57	biimtrid 242	. . . . 5 ⊢ (𝐴 ⊆ P → (𝑥 ∈ ∪ 𝐴 → ∃𝑦 ∈ ∪ 𝐴𝑥 <Q 𝑦))
59	46, 58	jcad 512	. . . 4 ⊢ (𝐴 ⊆ P → (𝑥 ∈ ∪ 𝐴 → (∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ ∪ 𝐴) ∧ ∃𝑦 ∈ ∪ 𝐴𝑥 <Q 𝑦)))
60	59	ralrimiv 3143	. . 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 11025	. 2 ⊢ (∪ 𝐴 ∈ P ↔ ((∅ ⊊ ∪ 𝐴 ∧ ∪ 𝐴 ⊊ Q) ∧ ∀𝑥 ∈ ∪ 𝐴(∀𝑦(𝑦 <Q 𝑥 → 𝑦 ∈ ∪ 𝐴) ∧ ∃𝑦 ∈ ∪ 𝐴𝑥 <Q 𝑦)))
63	35, 61, 62	sylanbrc 583	1 ⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥) → ∪ 𝐴 ∈ P)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1535  ∃wex 1776   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ∃wrex 3068   ⊆ wss 3963   ⊊ wpss 3964  ∅c0 4339  ∪ cuni 4912   class class class wbr 5148  Qcnq 10890   <_Q cltq 10896  Pcnp 10897  <_P cltp 10901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-om 7888  df-ni 10910  df-nq 10950  df-ltnq 10956  df-np 11019  df-ltp 11023

This theorem is referenced by:  supexpr  11092