Theorem suplem2pr 9875

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 9820	. . . . . 6 ⊢ <P ⊆ (P × P)
2	1	brel 5168	. . . . 5 ⊢ (𝑦<P ∪ 𝐴 → (𝑦 ∈ P ∧ ∪ 𝐴 ∈ P))
3	2	simpld 475	. . . 4 ⊢ (𝑦<P ∪ 𝐴 → 𝑦 ∈ P)
4		ralnex 2992	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 ¬ 𝑦<P 𝑧 ↔ ¬ ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)
5		ssel2 3598	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ P ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ P)
6		ltsopr 9854	. . . . . . . . . . . . . . . 16 ⊢ <P Or P
7		sotric 5061	. . . . . . . . . . . . . . . 16 ⊢ ((<P Or P ∧ (𝑦 ∈ P ∧ 𝑧 ∈ P)) → (𝑦<P 𝑧 ↔ ¬ (𝑦 = 𝑧 ∨ 𝑧<P 𝑦)))
8	6, 7	mpan 706	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦<P 𝑧 ↔ ¬ (𝑦 = 𝑧 ∨ 𝑧<P 𝑦)))
9	8	con2bid 344	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → ((𝑦 = 𝑧 ∨ 𝑧<P 𝑦) ↔ ¬ 𝑦<P 𝑧))
10	9	ancoms 469	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → ((𝑦 = 𝑧 ∨ 𝑧<P 𝑦) ↔ ¬ 𝑦<P 𝑧))
11		ltprord 9852	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → (𝑧<P 𝑦 ↔ 𝑧 ⊊ 𝑦))
12	11	orbi2d 738	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → ((𝑦 = 𝑧 ∨ 𝑧<P 𝑦) ↔ (𝑦 = 𝑧 ∨ 𝑧 ⊊ 𝑦)))
13		sspss 3706	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ⊆ 𝑦 ↔ (𝑧 ⊊ 𝑦 ∨ 𝑧 = 𝑦))
14		equcom 1945	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑦 ↔ 𝑦 = 𝑧)
15	14	orbi2i 541	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ⊊ 𝑦 ∨ 𝑧 = 𝑦) ↔ (𝑧 ⊊ 𝑦 ∨ 𝑦 = 𝑧))
16		orcom 402	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ⊊ 𝑦 ∨ 𝑦 = 𝑧) ↔ (𝑦 = 𝑧 ∨ 𝑧 ⊊ 𝑦))
17	13, 15, 16	3bitri 286	. . . . . . . . . . . . . 14 ⊢ (𝑧 ⊆ 𝑦 ↔ (𝑦 = 𝑧 ∨ 𝑧 ⊊ 𝑦))
18	12, 17	syl6bbr 278	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → ((𝑦 = 𝑧 ∨ 𝑧<P 𝑦) ↔ 𝑧 ⊆ 𝑦))
19	10, 18	bitr3d 270	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → (¬ 𝑦<P 𝑧 ↔ 𝑧 ⊆ 𝑦))
20	5, 19	sylan 488	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ P ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ P) → (¬ 𝑦<P 𝑧 ↔ 𝑧 ⊆ 𝑦))
21	20	an32s 846	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ P ∧ 𝑦 ∈ P) ∧ 𝑧 ∈ 𝐴) → (¬ 𝑦<P 𝑧 ↔ 𝑧 ⊆ 𝑦))
22	21	ralbidva 2985	. . . . . . . . 9 ⊢ ((𝐴 ⊆ P ∧ 𝑦 ∈ P) → (∀𝑧 ∈ 𝐴 ¬ 𝑦<P 𝑧 ↔ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝑦))
23	4, 22	syl5bbr 274	. . . . . . . 8 ⊢ ((𝐴 ⊆ P ∧ 𝑦 ∈ P) → (¬ ∃𝑧 ∈ 𝐴 𝑦<P 𝑧 ↔ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝑦))
24		unissb 4469	. . . . . . . 8 ⊢ (∪ 𝐴 ⊆ 𝑦 ↔ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝑦)
25	23, 24	syl6bbr 278	. . . . . . 7 ⊢ ((𝐴 ⊆ P ∧ 𝑦 ∈ P) → (¬ ∃𝑧 ∈ 𝐴 𝑦<P 𝑧 ↔ ∪ 𝐴 ⊆ 𝑦))
26		ssnpss 3710	. . . . . . . 8 ⊢ (∪ 𝐴 ⊆ 𝑦 → ¬ 𝑦 ⊊ ∪ 𝐴)
27		ltprord 9852	. . . . . . . . . 10 ⊢ ((𝑦 ∈ P ∧ ∪ 𝐴 ∈ P) → (𝑦<P ∪ 𝐴 ↔ 𝑦 ⊊ ∪ 𝐴))
28	27	biimpd 219	. . . . . . . . 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 135	. . . . . . 7 ⊢ (∪ 𝐴 ⊆ 𝑦 → ¬ 𝑦<P ∪ 𝐴)
31	25, 30	syl6bi 243	. . . . . 6 ⊢ ((𝐴 ⊆ P ∧ 𝑦 ∈ P) → (¬ ∃𝑧 ∈ 𝐴 𝑦<P 𝑧 → ¬ 𝑦<P ∪ 𝐴))
32	31	con4d 114	. . . . 5 ⊢ ((𝐴 ⊆ P ∧ 𝑦 ∈ P) → (𝑦<P ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))
33	32	ex 450	. . . 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 4467	. . . 4 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ⊆ ∪ 𝐴)
37		ssnpss 3710	. . . 4 ⊢ (𝑦 ⊆ ∪ 𝐴 → ¬ ∪ 𝐴 ⊊ 𝑦)
38	36, 37	syl 17	. . 3 ⊢ (𝑦 ∈ 𝐴 → ¬ ∪ 𝐴 ⊊ 𝑦)
39	1	brel 5168	. . . 4 ⊢ (∪ 𝐴<P 𝑦 → (∪ 𝐴 ∈ P ∧ 𝑦 ∈ P))
40		ltprord 9852	. . . . 5 ⊢ ((∪ 𝐴 ∈ P ∧ 𝑦 ∈ P) → (∪ 𝐴<P 𝑦 ↔ ∪ 𝐴 ⊊ 𝑦))
41	40	biimpd 219	. . . 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 135	. 2 ⊢ (𝑦 ∈ 𝐴 → ¬ ∪ 𝐴<P 𝑦)
44	35, 43	jctil 560	1 ⊢ (𝐴 ⊆ P → ((𝑦 ∈ 𝐴 → ¬ ∪ 𝐴<P 𝑦) ∧ (𝑦<P ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574   ⊊ wpss 3575  ∪ cuni 4436   class class class wbr 4653   Or wor 5034  Pcnp 9681  <_P cltp 9685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-omul 7565  df-er 7742  df-ni 9694  df-mi 9696  df-lti 9697  df-ltpq 9732  df-enq 9733  df-nq 9734  df-ltnq 9740  df-np 9803  df-ltp 9807

This theorem is referenced by:  supexpr  9876