Theorem suplocexprlemub 8054

Description: Lemma for suplocexpr 8056. 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 110	. . . . 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 8053	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ P)
7	6	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → 𝐵 ∈ P)
8	2, 3, 4	suplocexprlemss 8046	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ P)
9	8	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → 𝐴 ⊆ P)
10		simplr 529	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → 𝑦 ∈ 𝐴)
11	9, 10	sseldd 3243	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → 𝑦 ∈ P)
12		ltdfpr 7837	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ P) → (𝐵<P 𝑦 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦))))
13	7, 11, 12	syl2anc 411	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → (𝐵<P 𝑦 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦))))
14	1, 13	mpbid 147	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))
15		simprrl 541	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) → 𝑠 ∈ (2nd ‘𝐵))
16	5	suplocexprlem2b 8045	. . . . . . . . . . 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 2304	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (2nd ‘𝐵) ↔ 𝑠 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}))
19	18	ad3antrrr 492	. . . . . . . 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 147	. . . . . . 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 4118	. . . . . . . . 9 ⊢ (𝑢 = 𝑠 → (𝑤 <Q 𝑢 ↔ 𝑤 <Q 𝑠))
22	21	rexbidv 2545	. . . . . . . 8 ⊢ (𝑢 = 𝑠 → (∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑠))
23	22	elrab 2976	. . . . . . 7 ⊢ (𝑠 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ↔ (𝑠 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑠))
24	20, 23	sylib 122	. . . . . 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 114	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) → ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑠)
26		simprrr 542	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) → 𝑠 ∈ (1st ‘𝑦))
27	26	adantr 276	. . . . . . 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 533	. . . . . . . 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 488	. . . . . . . . . 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 7806	. . . . . . . . . 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 2298	. . . . . . . . . 10 ⊢ (𝑡 = (2nd ‘𝑦) → (𝑤 ∈ 𝑡 ↔ 𝑤 ∈ (2nd ‘𝑦)))
33		simprl 531	. . . . . . . . . . 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 2818	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
35	34	elint2 3961	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ∩ (2nd “ 𝐴) ↔ ∀𝑡 ∈ (2nd “ 𝐴)𝑤 ∈ 𝑡)
36	33, 35	sylib 122	. . . . . . . . . 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 6365	. . . . . . . . . . . . 13 ⊢ 2nd :V–onto→V
38		fofun 5596	. . . . . . . . . . . . 13 ⊢ (2nd :V–onto→V → Fun 2nd )
39	37, 38	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 2nd
40		vex 2818	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
41		fof 5595	. . . . . . . . . . . . . . 15 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
42	37, 41	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 2nd :V⟶V
43	42	fdmi 5521	. . . . . . . . . . . . 13 ⊢ dom 2nd = V
44	40, 43	eleqtrri 2310	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ dom 2nd
45		funfvima 5923	. . . . . . . . . . . 12 ⊢ ((Fun 2nd ∧ 𝑦 ∈ dom 2nd ) → (𝑦 ∈ 𝐴 → (2nd ‘𝑦) ∈ (2nd “ 𝐴)))
46	39, 44, 45	mp2an 426	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → (2nd ‘𝑦) ∈ (2nd “ 𝐴))
47	46	ad4antlr 495	. . . . . . . . . 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 2928	. . . . . . . . 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 7816	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ P ∧ 𝑤 ∈ (2nd ‘𝑦)) → (𝑤 <Q 𝑠 → 𝑠 ∈ (2nd ‘𝑦)))
50	31, 48, 49	syl2anc 411	. . . . . . . 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 306	. . . . . 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 537	. . . . . . 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 7823	. . . . . . 7 ⊢ ((⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ P ∧ 𝑠 ∈ Q) → ¬ (𝑠 ∈ (1st ‘𝑦) ∧ 𝑠 ∈ (2nd ‘𝑦)))
55	31, 53, 54	syl2anc 411	. . . . . 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 1418	. . . . 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 2667	. . . 4 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) → ⊥)
58	14, 57	rexlimddv 2667	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → ⊥)
59	58	inegd 1417	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ 𝐵<P 𝑦)
60	59	ralrimiva 2617	1 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ 𝐵<P 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   = wceq 1398  ⊥wfal 1403  ∃wex 1541   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  {crab 2526  Vcvv 2815   ⊆ wss 3214  ⟨cop 3697  ∪ cuni 3919  ∩ cint 3954   class class class wbr 4114  dom cdm 4754   “ cima 4757  Fun wfun 5351  ⟶wf 5353  –onto→wfo 5355  ‘cfv 5357  1^st c1st 6345  2^nd c2nd 6346  Qcnq 7611   <_Q cltq 7616  Pcnp 7622  <_P cltp 7626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-2o 6661  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684  df-enq0 7755  df-nq0 7756  df-0nq0 7757  df-plq0 7758  df-mq0 7759  df-inp 7797  df-iltp 7801

This theorem is referenced by:  suplocexpr  8056