Theorem suplocexprlemub 7685

Description: Lemma for suplocexpr 7687. The putative supremum is an upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)

Hypotheses

Ref	Expression
suplocexpr.m	⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
suplocexpr.ub	⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
suplocexpr.loc	⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
suplocexpr.b	⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩

Assertion

Ref	Expression
suplocexprlemub	⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ 𝐵<P 𝑦)

Distinct variable groups:   𝑢,𝐴,𝑤,𝑦   𝑥,𝐴,𝑧,𝑢,𝑦   𝑤,𝐵   𝜑,𝑢,𝑤,𝑦   𝜑,𝑥,𝑧   𝑧,𝑤

Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧,𝑢)

Proof of Theorem suplocexprlemub

Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 109	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → 𝐵<P 𝑦)
2		suplocexpr.m	. . . . . . . 8 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
3		suplocexpr.ub	. . . . . . . 8 ⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
4		suplocexpr.loc	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
5		suplocexpr.b	. . . . . . . 8 ⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩
6	2, 3, 4, 5	suplocexprlemex 7684	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ P)
7	6	ad2antrr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → 𝐵 ∈ P)
8	2, 3, 4	suplocexprlemss 7677	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ P)
9	8	ad2antrr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → 𝐴 ⊆ P)
10		simplr 525	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → 𝑦 ∈ 𝐴)
11	9, 10	sseldd 3148	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → 𝑦 ∈ P)
12		ltdfpr 7468	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ P) → (𝐵<P 𝑦 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦))))
13	7, 11, 12	syl2anc 409	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → (𝐵<P 𝑦 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦))))
14	1, 13	mpbid 146	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))
15		simprrl 534	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) → 𝑠 ∈ (2nd ‘𝐵))
16	5	suplocexprlem2b 7676	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ P → (2nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
17	8, 16	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → (2nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
18	17	eleq2d 2240	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (2nd ‘𝐵) ↔ 𝑠 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}))
19	18	ad3antrrr 489	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) → (𝑠 ∈ (2nd ‘𝐵) ↔ 𝑠 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}))
20	15, 19	mpbid 146	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) → 𝑠 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
21		breq2 3993	. . . . . . . . 9 ⊢ (𝑢 = 𝑠 → (𝑤 <Q 𝑢 ↔ 𝑤 <Q 𝑠))
22	21	rexbidv 2471	. . . . . . . 8 ⊢ (𝑢 = 𝑠 → (∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑠))
23	22	elrab 2886	. . . . . . 7 ⊢ (𝑠 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ↔ (𝑠 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑠))
24	20, 23	sylib 121	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) → (𝑠 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑠))
25	24	simprd 113	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) → ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑠)
26		simprrr 535	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) → 𝑠 ∈ (1st ‘𝑦))
27	26	adantr 274	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑠)) → 𝑠 ∈ (1st ‘𝑦))
28		simprr 527	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑠)) → 𝑤 <Q 𝑠)
29	11	ad2antrr 485	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑠)) → 𝑦 ∈ P)
30		prop 7437	. . . . . . . . . 10 ⊢ (𝑦 ∈ P → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ P)
31	29, 30	syl 14	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑠)) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ P)
32		eleq2 2234	. . . . . . . . . 10 ⊢ (𝑡 = (2nd ‘𝑦) → (𝑤 ∈ 𝑡 ↔ 𝑤 ∈ (2nd ‘𝑦)))
33		simprl 526	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑠)) → 𝑤 ∈ ∩ (2nd “ 𝐴))
34		vex 2733	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
35	34	elint2 3838	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ∩ (2nd “ 𝐴) ↔ ∀𝑡 ∈ (2nd “ 𝐴)𝑤 ∈ 𝑡)
36	33, 35	sylib 121	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑠)) → ∀𝑡 ∈ (2nd “ 𝐴)𝑤 ∈ 𝑡)
37		fo2nd 6137	. . . . . . . . . . . . 13 ⊢ 2nd :V–onto→V
38		fofun 5421	. . . . . . . . . . . . 13 ⊢ (2nd :V–onto→V → Fun 2nd )
39	37, 38	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 2nd
40		vex 2733	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
41		fof 5420	. . . . . . . . . . . . . . 15 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
42	37, 41	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 2nd :V⟶V
43	42	fdmi 5355	. . . . . . . . . . . . 13 ⊢ dom 2nd = V
44	40, 43	eleqtrri 2246	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ dom 2nd
45		funfvima 5727	. . . . . . . . . . . 12 ⊢ ((Fun 2nd ∧ 𝑦 ∈ dom 2nd ) → (𝑦 ∈ 𝐴 → (2nd ‘𝑦) ∈ (2nd “ 𝐴)))
46	39, 44, 45	mp2an 424	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → (2nd ‘𝑦) ∈ (2nd “ 𝐴))
47	46	ad4antlr 492	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑠)) → (2nd ‘𝑦) ∈ (2nd “ 𝐴))
48	32, 36, 47	rspcdva 2839	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑠)) → 𝑤 ∈ (2nd ‘𝑦))
49		prcunqu 7447	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ P ∧ 𝑤 ∈ (2nd ‘𝑦)) → (𝑤 <Q 𝑠 → 𝑠 ∈ (2nd ‘𝑦)))
50	31, 48, 49	syl2anc 409	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑠)) → (𝑤 <Q 𝑠 → 𝑠 ∈ (2nd ‘𝑦)))
51	28, 50	mpd 13	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑠)) → 𝑠 ∈ (2nd ‘𝑦))
52	27, 51	jca 304	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑠)) → (𝑠 ∈ (1st ‘𝑦) ∧ 𝑠 ∈ (2nd ‘𝑦)))
53		simplrl 530	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑠)) → 𝑠 ∈ Q)
54		prdisj 7454	. . . . . . 7 ⊢ ((⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ P ∧ 𝑠 ∈ Q) → ¬ (𝑠 ∈ (1st ‘𝑦) ∧ 𝑠 ∈ (2nd ‘𝑦)))
55	31, 53, 54	syl2anc 409	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑠)) → ¬ (𝑠 ∈ (1st ‘𝑦) ∧ 𝑠 ∈ (2nd ‘𝑦)))
56	52, 55	pm2.21fal 1368	. . . . 5 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑠)) → ⊥)
57	25, 56	rexlimddv 2592	. . . 4 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) → ⊥)
58	14, 57	rexlimddv 2592	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → ⊥)
59	58	inegd 1367	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ 𝐵<P 𝑦)
60	59	ralrimiva 2543	1 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ 𝐵<P 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 703   = wceq 1348  ⊥wfal 1353  ∃wex 1485   ∈ wcel 2141  ∀wral 2448  ∃wrex 2449  {crab 2452  Vcvv 2730   ⊆ wss 3121  ⟨cop 3586  ∪ cuni 3796  ∩ cint 3831   class class class wbr 3989  dom cdm 4611   “ cima 4614  Fun wfun 5192  ⟶wf 5194  –onto→wfo 5196  ‘cfv 5198  1^st c1st 6117  2^nd c2nd 6118  Qcnq 7242   <_Q cltq 7247  Pcnp 7253  <_P cltp 7257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iltp 7432

This theorem is referenced by:  suplocexpr  7687