Theorem suplem2pr 11030

Description: The union of a set of positive reals (if a positive real) is its supremum (the least upper bound). 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
suplem2pr	⊢ (𝐴 ⊆ P → ((𝑦 ∈ 𝐴 → ¬ ∪ 𝐴<P 𝑦) ∧ (𝑦<P ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)))

Distinct variable group:   𝑦,𝑧,𝐴

Proof of Theorem suplem2pr

Step	Hyp	Ref	Expression
1		ltrelpr 10975	. . . . . 6 ⊢ <P ⊆ (P × P)
2	1	brel 5733	. . . . 5 ⊢ (𝑦<P ∪ 𝐴 → (𝑦 ∈ P ∧ ∪ 𝐴 ∈ P))
3	2	simpld 495	. . . 4 ⊢ (𝑦<P ∪ 𝐴 → 𝑦 ∈ P)
4		ralnex 3071	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 ¬ 𝑦<P 𝑧 ↔ ¬ ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)
5		ssel2 3973	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ P ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ P)
6		ltsopr 11009	. . . . . . . . . . . . . . . 16 ⊢ <P Or P
7		sotric 5609	. . . . . . . . . . . . . . . 16 ⊢ ((<P Or P ∧ (𝑦 ∈ P ∧ 𝑧 ∈ P)) → (𝑦<P 𝑧 ↔ ¬ (𝑦 = 𝑧 ∨ 𝑧<P 𝑦)))
8	6, 7	mpan 688	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦<P 𝑧 ↔ ¬ (𝑦 = 𝑧 ∨ 𝑧<P 𝑦)))
9	8	con2bid 354	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → ((𝑦 = 𝑧 ∨ 𝑧<P 𝑦) ↔ ¬ 𝑦<P 𝑧))
10	9	ancoms 459	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → ((𝑦 = 𝑧 ∨ 𝑧<P 𝑦) ↔ ¬ 𝑦<P 𝑧))
11		ltprord 11007	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → (𝑧<P 𝑦 ↔ 𝑧 ⊊ 𝑦))
12	11	orbi2d 914	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → ((𝑦 = 𝑧 ∨ 𝑧<P 𝑦) ↔ (𝑦 = 𝑧 ∨ 𝑧 ⊊ 𝑦)))
13		sspss 4095	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ⊆ 𝑦 ↔ (𝑧 ⊊ 𝑦 ∨ 𝑧 = 𝑦))
14		equcom 2021	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑦 ↔ 𝑦 = 𝑧)
15	14	orbi2i 911	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ⊊ 𝑦 ∨ 𝑧 = 𝑦) ↔ (𝑧 ⊊ 𝑦 ∨ 𝑦 = 𝑧))
16		orcom 868	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ⊊ 𝑦 ∨ 𝑦 = 𝑧) ↔ (𝑦 = 𝑧 ∨ 𝑧 ⊊ 𝑦))
17	13, 15, 16	3bitri 296	. . . . . . . . . . . . . 14 ⊢ (𝑧 ⊆ 𝑦 ↔ (𝑦 = 𝑧 ∨ 𝑧 ⊊ 𝑦))
18	12, 17	bitr4di 288	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → ((𝑦 = 𝑧 ∨ 𝑧<P 𝑦) ↔ 𝑧 ⊆ 𝑦))
19	10, 18	bitr3d 280	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → (¬ 𝑦<P 𝑧 ↔ 𝑧 ⊆ 𝑦))
20	5, 19	sylan 580	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ P ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ P) → (¬ 𝑦<P 𝑧 ↔ 𝑧 ⊆ 𝑦))
21	20	an32s 650	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ P ∧ 𝑦 ∈ P) ∧ 𝑧 ∈ 𝐴) → (¬ 𝑦<P 𝑧 ↔ 𝑧 ⊆ 𝑦))
22	21	ralbidva 3174	. . . . . . . . 9 ⊢ ((𝐴 ⊆ P ∧ 𝑦 ∈ P) → (∀𝑧 ∈ 𝐴 ¬ 𝑦<P 𝑧 ↔ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝑦))
23	4, 22	bitr3id 284	. . . . . . . 8 ⊢ ((𝐴 ⊆ P ∧ 𝑦 ∈ P) → (¬ ∃𝑧 ∈ 𝐴 𝑦<P 𝑧 ↔ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝑦))
24		unissb 4936	. . . . . . . 8 ⊢ (∪ 𝐴 ⊆ 𝑦 ↔ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝑦)
25	23, 24	bitr4di 288	. . . . . . 7 ⊢ ((𝐴 ⊆ P ∧ 𝑦 ∈ P) → (¬ ∃𝑧 ∈ 𝐴 𝑦<P 𝑧 ↔ ∪ 𝐴 ⊆ 𝑦))
26		ssnpss 4099	. . . . . . . 8 ⊢ (∪ 𝐴 ⊆ 𝑦 → ¬ 𝑦 ⊊ ∪ 𝐴)
27		ltprord 11007	. . . . . . . . . 10 ⊢ ((𝑦 ∈ P ∧ ∪ 𝐴 ∈ P) → (𝑦<P ∪ 𝐴 ↔ 𝑦 ⊊ ∪ 𝐴))
28	27	biimpd 228	. . . . . . . . 9 ⊢ ((𝑦 ∈ P ∧ ∪ 𝐴 ∈ P) → (𝑦<P ∪ 𝐴 → 𝑦 ⊊ ∪ 𝐴))
29	2, 28	mpcom 38	. . . . . . . 8 ⊢ (𝑦<P ∪ 𝐴 → 𝑦 ⊊ ∪ 𝐴)
30	26, 29	nsyl 140	. . . . . . 7 ⊢ (∪ 𝐴 ⊆ 𝑦 → ¬ 𝑦<P ∪ 𝐴)
31	25, 30	syl6bi 252	. . . . . 6 ⊢ ((𝐴 ⊆ P ∧ 𝑦 ∈ P) → (¬ ∃𝑧 ∈ 𝐴 𝑦<P 𝑧 → ¬ 𝑦<P ∪ 𝐴))
32	31	con4d 115	. . . . 5 ⊢ ((𝐴 ⊆ P ∧ 𝑦 ∈ P) → (𝑦<P ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))
33	32	ex 413	. . . 4 ⊢ (𝐴 ⊆ P → (𝑦 ∈ P → (𝑦<P ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)))
34	3, 33	syl5 34	. . 3 ⊢ (𝐴 ⊆ P → (𝑦<P ∪ 𝐴 → (𝑦<P ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)))
35	34	pm2.43d 53	. 2 ⊢ (𝐴 ⊆ P → (𝑦<P ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))
36		elssuni 4934	. . . 4 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ⊆ ∪ 𝐴)
37		ssnpss 4099	. . . 4 ⊢ (𝑦 ⊆ ∪ 𝐴 → ¬ ∪ 𝐴 ⊊ 𝑦)
38	36, 37	syl 17	. . 3 ⊢ (𝑦 ∈ 𝐴 → ¬ ∪ 𝐴 ⊊ 𝑦)
39	1	brel 5733	. . . 4 ⊢ (∪ 𝐴<P 𝑦 → (∪ 𝐴 ∈ P ∧ 𝑦 ∈ P))
40		ltprord 11007	. . . . 5 ⊢ ((∪ 𝐴 ∈ P ∧ 𝑦 ∈ P) → (∪ 𝐴<P 𝑦 ↔ ∪ 𝐴 ⊊ 𝑦))
41	40	biimpd 228	. . . 4 ⊢ ((∪ 𝐴 ∈ P ∧ 𝑦 ∈ P) → (∪ 𝐴<P 𝑦 → ∪ 𝐴 ⊊ 𝑦))
42	39, 41	mpcom 38	. . 3 ⊢ (∪ 𝐴<P 𝑦 → ∪ 𝐴 ⊊ 𝑦)
43	38, 42	nsyl 140	. 2 ⊢ (𝑦 ∈ 𝐴 → ¬ ∪ 𝐴<P 𝑦)
44	35, 43	jctil 520	1 ⊢ (𝐴 ⊆ P → ((𝑦 ∈ 𝐴 → ¬ ∪ 𝐴<P 𝑦) ∧ (𝑦<P ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∈ wcel 2106  ∀wral 3060  ∃wrex 3069   ⊆ wss 3944   ⊊ wpss 3945  ∪ cuni 4901   class class class wbr 5141   Or wor 5580  Pcnp 10836  <_P cltp 10840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-oadd 8452  df-omul 8453  df-er 8686  df-ni 10849  df-mi 10851  df-lti 10852  df-ltpq 10887  df-enq 10888  df-nq 10889  df-ltnq 10895  df-np 10958  df-ltp 10962

This theorem is referenced by:  supexpr  11031