Theorem suplocexprlemub 8080

Description: Lemma for suplocexpr 8082. 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 8079	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ P)
7	6	ad2antrr 492	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → 𝐵 ∈ P)
8	2, 3, 4	suplocexprlemss 8072	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ P)
9	8	ad2antrr 492	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → 𝐴 ⊆ P)
10		simplr 533	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → 𝑦 ∈ 𝐴)
11	9, 10	sseldd 3249	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → 𝑦 ∈ P)
12		ltdfpr 7863	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ P) → (𝐵<P 𝑦 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦))))
13	7, 11, 12	syl2anc 415	. . . . 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 545	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) → 𝑠 ∈ (2nd ‘𝐵))
16	5	suplocexprlem2b 8071	. . . . . . . . . . 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 2308	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (2nd ‘𝐵) ↔ 𝑠 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}))
19	18	ad3antrrr 496	. . . . . . . 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 4129	. . . . . . . . 9 ⊢ (𝑢 = 𝑠 → (𝑤 <Q 𝑢 ↔ 𝑤 <Q 𝑠))
22	21	rexbidv 2551	. . . . . . . 8 ⊢ (𝑢 = 𝑠 → (∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑠))
23	22	elrab 2982	. . . . . . 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 546	. . . . . . . 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 537	. . . . . . . 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 492	. . . . . . . . . 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 7832	. . . . . . . . . 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 2302	. . . . . . . . . 10 ⊢ (𝑡 = (2nd ‘𝑦) → (𝑤 ∈ 𝑡 ↔ 𝑤 ∈ (2nd ‘𝑦)))
33		simprl 535	. . . . . . . . . . 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 2824	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
35	34	elint2 3972	. . . . . . . . . . 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 6382	. . . . . . . . . . . . 13 ⊢ 2nd :V–onto→V
38		fofun 5611	. . . . . . . . . . . . 13 ⊢ (2nd :V–onto→V → Fun 2nd )
39	37, 38	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 2nd
40		vex 2824	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
41		fof 5610	. . . . . . . . . . . . . . 15 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
42	37, 41	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 2nd :V⟶V
43	42	fdmi 5536	. . . . . . . . . . . . 13 ⊢ dom 2nd = V
44	40, 43	eleqtrri 2314	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ dom 2nd
45		funfvima 5940	. . . . . . . . . . . 12 ⊢ ((Fun 2nd ∧ 𝑦 ∈ dom 2nd ) → (𝑦 ∈ 𝐴 → (2nd ‘𝑦) ∈ (2nd “ 𝐴)))
46	39, 44, 45	mp2an 430	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → (2nd ‘𝑦) ∈ (2nd “ 𝐴))
47	46	ad4antlr 499	. . . . . . . . . 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 2934	. . . . . . . . 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 7842	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ P ∧ 𝑤 ∈ (2nd ‘𝑦)) → (𝑤 <Q 𝑠 → 𝑠 ∈ (2nd ‘𝑦)))
50	31, 48, 49	syl2anc 415	. . . . . . . 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 541	. . . . . . 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 7849	. . . . . . 7 ⊢ ((⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ P ∧ 𝑠 ∈ Q) → ¬ (𝑠 ∈ (1st ‘𝑦) ∧ 𝑠 ∈ (2nd ‘𝑦)))
55	31, 53, 54	syl2anc 415	. . . . . 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 1422	. . . . 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 2673	. . . 4 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) → ⊥)
58	14, 57	rexlimddv 2673	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → ⊥)
59	58	inegd 1421	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ 𝐵<P 𝑦)
60	59	ralrimiva 2623	1 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ 𝐵<P 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   = wceq 1402  ⊥wfal 1407  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  {crab 2532  Vcvv 2821   ⊆ wss 3220  ⟨cop 3708  ∪ cuni 3930  ∩ cint 3965   class class class wbr 4125  dom cdm 4769   “ cima 4772  Fun wfun 5366  ⟶wf 5368  –onto→wfo 5370  ‘cfv 5372  1^st c1st 6362  2^nd c2nd 6363  Qcnq 7637   <_Q cltq 7642  Pcnp 7648  <_P cltp 7652

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-2o 6678  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-enq0 7781  df-nq0 7782  df-0nq0 7783  df-plq0 7784  df-mq0 7785  df-inp 7823  df-iltp 7827

This theorem is referenced by:  suplocexpr  8082