Theorem suplem2pr 11048

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 10993	. . . . . 6 ⊢ <P ⊆ (P × P)
2	1	brel 5742	. . . . 5 ⊢ (𝑦<P ∪ 𝐴 → (𝑦 ∈ P ∧ ∪ 𝐴 ∈ P))
3	2	simpld 496	. . . 4 ⊢ (𝑦<P ∪ 𝐴 → 𝑦 ∈ P)
4		ralnex 3073	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 ¬ 𝑦<P 𝑧 ↔ ¬ ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)
5		ssel2 3978	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ P ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ P)
6		ltsopr 11027	. . . . . . . . . . . . . . . 16 ⊢ <P Or P
7		sotric 5617	. . . . . . . . . . . . . . . 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 355	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → ((𝑦 = 𝑧 ∨ 𝑧<P 𝑦) ↔ ¬ 𝑦<P 𝑧))
10	9	ancoms 460	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → ((𝑦 = 𝑧 ∨ 𝑧<P 𝑦) ↔ ¬ 𝑦<P 𝑧))
11		ltprord 11025	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → (𝑧<P 𝑦 ↔ 𝑧 ⊊ 𝑦))
12	11	orbi2d 915	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → ((𝑦 = 𝑧 ∨ 𝑧<P 𝑦) ↔ (𝑦 = 𝑧 ∨ 𝑧 ⊊ 𝑦)))
13		sspss 4100	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ⊆ 𝑦 ↔ (𝑧 ⊊ 𝑦 ∨ 𝑧 = 𝑦))
14		equcom 2022	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑦 ↔ 𝑦 = 𝑧)
15	14	orbi2i 912	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ⊊ 𝑦 ∨ 𝑧 = 𝑦) ↔ (𝑧 ⊊ 𝑦 ∨ 𝑦 = 𝑧))
16		orcom 869	. . . . . . . . . . . . . . 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 651	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ P ∧ 𝑦 ∈ P) ∧ 𝑧 ∈ 𝐴) → (¬ 𝑦<P 𝑧 ↔ 𝑧 ⊆ 𝑦))
22	21	ralbidva 3176	. . . . . . . . 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 4944	. . . . . . . 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 4104	. . . . . . . 8 ⊢ (∪ 𝐴 ⊆ 𝑦 → ¬ 𝑦 ⊊ ∪ 𝐴)
27		ltprord 11025	. . . . . . . . . 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 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 414	. . . 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 4942	. . . 4 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ⊆ ∪ 𝐴)
37		ssnpss 4104	. . . 4 ⊢ (𝑦 ⊆ ∪ 𝐴 → ¬ ∪ 𝐴 ⊊ 𝑦)
38	36, 37	syl 17	. . 3 ⊢ (𝑦 ∈ 𝐴 → ¬ ∪ 𝐴 ⊊ 𝑦)
39	1	brel 5742	. . . 4 ⊢ (∪ 𝐴<P 𝑦 → (∪ 𝐴 ∈ P ∧ 𝑦 ∈ P))
40		ltprord 11025	. . . . 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 521	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 397   ∨ wo 846   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071   ⊆ wss 3949   ⊊ wpss 3950  ∪ cuni 4909   class class class wbr 5149   Or wor 5588  Pcnp 10854  <_P cltp 10858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-oadd 8470  df-omul 8471  df-er 8703  df-ni 10867  df-mi 10869  df-lti 10870  df-ltpq 10905  df-enq 10906  df-nq 10907  df-ltnq 10913  df-np 10976  df-ltp 10980

This theorem is referenced by:  supexpr  11049