Theorem suplem2pr 10464

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 10409	. . . . . 6 ⊢ <P ⊆ (P × P)
2	1	brel 5581	. . . . 5 ⊢ (𝑦<P ∪ 𝐴 → (𝑦 ∈ P ∧ ∪ 𝐴 ∈ P))
3	2	simpld 498	. . . 4 ⊢ (𝑦<P ∪ 𝐴 → 𝑦 ∈ P)
4		ralnex 3199	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 ¬ 𝑦<P 𝑧 ↔ ¬ ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)
5		ssel2 3910	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ P ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ P)
6		ltsopr 10443	. . . . . . . . . . . . . . . 16 ⊢ <P Or P
7		sotric 5465	. . . . . . . . . . . . . . . 16 ⊢ ((<P Or P ∧ (𝑦 ∈ P ∧ 𝑧 ∈ P)) → (𝑦<P 𝑧 ↔ ¬ (𝑦 = 𝑧 ∨ 𝑧<P 𝑦)))
8	6, 7	mpan 689	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦<P 𝑧 ↔ ¬ (𝑦 = 𝑧 ∨ 𝑧<P 𝑦)))
9	8	con2bid 358	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → ((𝑦 = 𝑧 ∨ 𝑧<P 𝑦) ↔ ¬ 𝑦<P 𝑧))
10	9	ancoms 462	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → ((𝑦 = 𝑧 ∨ 𝑧<P 𝑦) ↔ ¬ 𝑦<P 𝑧))
11		ltprord 10441	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → (𝑧<P 𝑦 ↔ 𝑧 ⊊ 𝑦))
12	11	orbi2d 913	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → ((𝑦 = 𝑧 ∨ 𝑧<P 𝑦) ↔ (𝑦 = 𝑧 ∨ 𝑧 ⊊ 𝑦)))
13		sspss 4027	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ⊆ 𝑦 ↔ (𝑧 ⊊ 𝑦 ∨ 𝑧 = 𝑦))
14		equcom 2025	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑦 ↔ 𝑦 = 𝑧)
15	14	orbi2i 910	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ⊊ 𝑦 ∨ 𝑧 = 𝑦) ↔ (𝑧 ⊊ 𝑦 ∨ 𝑦 = 𝑧))
16		orcom 867	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ⊊ 𝑦 ∨ 𝑦 = 𝑧) ↔ (𝑦 = 𝑧 ∨ 𝑧 ⊊ 𝑦))
17	13, 15, 16	3bitri 300	. . . . . . . . . . . . . 14 ⊢ (𝑧 ⊆ 𝑦 ↔ (𝑦 = 𝑧 ∨ 𝑧 ⊊ 𝑦))
18	12, 17	syl6bbr 292	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → ((𝑦 = 𝑧 ∨ 𝑧<P 𝑦) ↔ 𝑧 ⊆ 𝑦))
19	10, 18	bitr3d 284	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → (¬ 𝑦<P 𝑧 ↔ 𝑧 ⊆ 𝑦))
20	5, 19	sylan 583	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ P ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ P) → (¬ 𝑦<P 𝑧 ↔ 𝑧 ⊆ 𝑦))
21	20	an32s 651	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ P ∧ 𝑦 ∈ P) ∧ 𝑧 ∈ 𝐴) → (¬ 𝑦<P 𝑧 ↔ 𝑧 ⊆ 𝑦))
22	21	ralbidva 3161	. . . . . . . . 9 ⊢ ((𝐴 ⊆ P ∧ 𝑦 ∈ P) → (∀𝑧 ∈ 𝐴 ¬ 𝑦<P 𝑧 ↔ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝑦))
23	4, 22	bitr3id 288	. . . . . . . 8 ⊢ ((𝐴 ⊆ P ∧ 𝑦 ∈ P) → (¬ ∃𝑧 ∈ 𝐴 𝑦<P 𝑧 ↔ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝑦))
24		unissb 4832	. . . . . . . 8 ⊢ (∪ 𝐴 ⊆ 𝑦 ↔ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝑦)
25	23, 24	syl6bbr 292	. . . . . . 7 ⊢ ((𝐴 ⊆ P ∧ 𝑦 ∈ P) → (¬ ∃𝑧 ∈ 𝐴 𝑦<P 𝑧 ↔ ∪ 𝐴 ⊆ 𝑦))
26		ssnpss 4031	. . . . . . . 8 ⊢ (∪ 𝐴 ⊆ 𝑦 → ¬ 𝑦 ⊊ ∪ 𝐴)
27		ltprord 10441	. . . . . . . . . 10 ⊢ ((𝑦 ∈ P ∧ ∪ 𝐴 ∈ P) → (𝑦<P ∪ 𝐴 ↔ 𝑦 ⊊ ∪ 𝐴))
28	27	biimpd 232	. . . . . . . . 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 142	. . . . . . 7 ⊢ (∪ 𝐴 ⊆ 𝑦 → ¬ 𝑦<P ∪ 𝐴)
31	25, 30	syl6bi 256	. . . . . 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 416	. . . 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 4830	. . . 4 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ⊆ ∪ 𝐴)
37		ssnpss 4031	. . . 4 ⊢ (𝑦 ⊆ ∪ 𝐴 → ¬ ∪ 𝐴 ⊊ 𝑦)
38	36, 37	syl 17	. . 3 ⊢ (𝑦 ∈ 𝐴 → ¬ ∪ 𝐴 ⊊ 𝑦)
39	1	brel 5581	. . . 4 ⊢ (∪ 𝐴<P 𝑦 → (∪ 𝐴 ∈ P ∧ 𝑦 ∈ P))
40		ltprord 10441	. . . . 5 ⊢ ((∪ 𝐴 ∈ P ∧ 𝑦 ∈ P) → (∪ 𝐴<P 𝑦 ↔ ∪ 𝐴 ⊊ 𝑦))
41	40	biimpd 232	. . . 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 142	. 2 ⊢ (𝑦 ∈ 𝐴 → ¬ ∪ 𝐴<P 𝑦)
44	35, 43	jctil 523	1 ⊢ (𝐴 ⊆ P → ((𝑦 ∈ 𝐴 → ¬ ∪ 𝐴<P 𝑦) ∧ (𝑦<P ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ⊆ wss 3881   ⊊ wpss 3882  ∪ cuni 4800   class class class wbr 5030   Or wor 5437  Pcnp 10270  <_P cltp 10274

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-oadd 8089  df-omul 8090  df-er 8272  df-ni 10283  df-mi 10285  df-lti 10286  df-ltpq 10321  df-enq 10322  df-nq 10323  df-ltnq 10329  df-np 10392  df-ltp 10396

This theorem is referenced by:  supexpr  10465