Theorem suplem2pr 10965

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 10910	. . . . . 6 ⊢ <P ⊆ (P × P)
2	1	brel 5687	. . . . 5 ⊢ (𝑦<P ∪ 𝐴 → (𝑦 ∈ P ∧ ∪ 𝐴 ∈ P))
3	2	simpld 494	. . . 4 ⊢ (𝑦<P ∪ 𝐴 → 𝑦 ∈ P)
4		ralnex 3064	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 ¬ 𝑦<P 𝑧 ↔ ¬ ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)
5		ssel2 3917	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ P ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ P)
6		ltsopr 10944	. . . . . . . . . . . . . . . 16 ⊢ <P Or P
7		sotric 5560	. . . . . . . . . . . . . . . 16 ⊢ ((<P Or P ∧ (𝑦 ∈ P ∧ 𝑧 ∈ P)) → (𝑦<P 𝑧 ↔ ¬ (𝑦 = 𝑧 ∨ 𝑧<P 𝑦)))
8	6, 7	mpan 691	. . . . . . . . . . . . . . 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 458	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → ((𝑦 = 𝑧 ∨ 𝑧<P 𝑦) ↔ ¬ 𝑦<P 𝑧))
11		ltprord 10942	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → (𝑧<P 𝑦 ↔ 𝑧 ⊊ 𝑦))
12	11	orbi2d 916	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → ((𝑦 = 𝑧 ∨ 𝑧<P 𝑦) ↔ (𝑦 = 𝑧 ∨ 𝑧 ⊊ 𝑦)))
13		sspss 4043	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ⊆ 𝑦 ↔ (𝑧 ⊊ 𝑦 ∨ 𝑧 = 𝑦))
14		equcom 2020	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑦 ↔ 𝑦 = 𝑧)
15	14	orbi2i 913	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ⊊ 𝑦 ∨ 𝑧 = 𝑦) ↔ (𝑧 ⊊ 𝑦 ∨ 𝑦 = 𝑧))
16		orcom 871	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ⊊ 𝑦 ∨ 𝑦 = 𝑧) ↔ (𝑦 = 𝑧 ∨ 𝑧 ⊊ 𝑦))
17	13, 15, 16	3bitri 297	. . . . . . . . . . . . . 14 ⊢ (𝑧 ⊆ 𝑦 ↔ (𝑦 = 𝑧 ∨ 𝑧 ⊊ 𝑦))
18	12, 17	bitr4di 289	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → ((𝑦 = 𝑧 ∨ 𝑧<P 𝑦) ↔ 𝑧 ⊆ 𝑦))
19	10, 18	bitr3d 281	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → (¬ 𝑦<P 𝑧 ↔ 𝑧 ⊆ 𝑦))
20	5, 19	sylan 581	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ P ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ P) → (¬ 𝑦<P 𝑧 ↔ 𝑧 ⊆ 𝑦))
21	20	an32s 653	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ P ∧ 𝑦 ∈ P) ∧ 𝑧 ∈ 𝐴) → (¬ 𝑦<P 𝑧 ↔ 𝑧 ⊆ 𝑦))
22	21	ralbidva 3159	. . . . . . . . 9 ⊢ ((𝐴 ⊆ P ∧ 𝑦 ∈ P) → (∀𝑧 ∈ 𝐴 ¬ 𝑦<P 𝑧 ↔ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝑦))
23	4, 22	bitr3id 285	. . . . . . . 8 ⊢ ((𝐴 ⊆ P ∧ 𝑦 ∈ P) → (¬ ∃𝑧 ∈ 𝐴 𝑦<P 𝑧 ↔ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝑦))
24		unissb 4884	. . . . . . . 8 ⊢ (∪ 𝐴 ⊆ 𝑦 ↔ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝑦)
25	23, 24	bitr4di 289	. . . . . . 7 ⊢ ((𝐴 ⊆ P ∧ 𝑦 ∈ P) → (¬ ∃𝑧 ∈ 𝐴 𝑦<P 𝑧 ↔ ∪ 𝐴 ⊆ 𝑦))
26		ssnpss 4047	. . . . . . . 8 ⊢ (∪ 𝐴 ⊆ 𝑦 → ¬ 𝑦 ⊊ ∪ 𝐴)
27		ltprord 10942	. . . . . . . . . 10 ⊢ ((𝑦 ∈ P ∧ ∪ 𝐴 ∈ P) → (𝑦<P ∪ 𝐴 ↔ 𝑦 ⊊ ∪ 𝐴))
28	27	biimpd 229	. . . . . . . . 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	biimtrdi 253	. . . . . 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 412	. . . 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 4882	. . . 4 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ⊆ ∪ 𝐴)
37		ssnpss 4047	. . . 4 ⊢ (𝑦 ⊆ ∪ 𝐴 → ¬ ∪ 𝐴 ⊊ 𝑦)
38	36, 37	syl 17	. . 3 ⊢ (𝑦 ∈ 𝐴 → ¬ ∪ 𝐴 ⊊ 𝑦)
39	1	brel 5687	. . . 4 ⊢ (∪ 𝐴<P 𝑦 → (∪ 𝐴 ∈ P ∧ 𝑦 ∈ P))
40		ltprord 10942	. . . . 5 ⊢ ((∪ 𝐴 ∈ P ∧ 𝑦 ∈ P) → (∪ 𝐴<P 𝑦 ↔ ∪ 𝐴 ⊊ 𝑦))
41	40	biimpd 229	. . . 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 519	1 ⊢ (𝐴 ⊆ P → ((𝑦 ∈ 𝐴 → ¬ ∪ 𝐴<P 𝑦) ∧ (𝑦<P ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ⊆ wss 3890   ⊊ wpss 3891  ∪ cuni 4851   class class class wbr 5086   Or wor 5529  Pcnp 10771  <_P cltp 10775

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-oadd 8400  df-omul 8401  df-er 8634  df-ni 10784  df-mi 10786  df-lti 10787  df-ltpq 10822  df-enq 10823  df-nq 10824  df-ltnq 10830  df-np 10893  df-ltp 10897

This theorem is referenced by:  supexpr  10966