Theorem suplocexprlemub 7724

Description: Lemma for suplocexpr 7726. 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 7723	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ P)
7	6	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → 𝐵 ∈ P)
8	2, 3, 4	suplocexprlemss 7716	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ P)
9	8	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → 𝐴 ⊆ P)
10		simplr 528	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → 𝑦 ∈ 𝐴)
11	9, 10	sseldd 3158	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → 𝑦 ∈ P)
12		ltdfpr 7507	. . . . . 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 539	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) → 𝑠 ∈ (2nd ‘𝐵))
16	5	suplocexprlem2b 7715	. . . . . . . . . . 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 2247	. . . . . . . . 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 4009	. . . . . . . . 9 ⊢ (𝑢 = 𝑠 → (𝑤 <Q 𝑢 ↔ 𝑤 <Q 𝑠))
22	21	rexbidv 2478	. . . . . . . 8 ⊢ (𝑢 = 𝑠 → (∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑠))
23	22	elrab 2895	. . . . . . 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 540	. . . . . . . 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 531	. . . . . . . 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 7476	. . . . . . . . . 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 2241	. . . . . . . . . 10 ⊢ (𝑡 = (2nd ‘𝑦) → (𝑤 ∈ 𝑡 ↔ 𝑤 ∈ (2nd ‘𝑦)))
33		simprl 529	. . . . . . . . . . 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 2742	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
35	34	elint2 3853	. . . . . . . . . . 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 6161	. . . . . . . . . . . . 13 ⊢ 2nd :V–onto→V
38		fofun 5441	. . . . . . . . . . . . 13 ⊢ (2nd :V–onto→V → Fun 2nd )
39	37, 38	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 2nd
40		vex 2742	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
41		fof 5440	. . . . . . . . . . . . . . 15 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
42	37, 41	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 2nd :V⟶V
43	42	fdmi 5375	. . . . . . . . . . . . 13 ⊢ dom 2nd = V
44	40, 43	eleqtrri 2253	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ dom 2nd
45		funfvima 5750	. . . . . . . . . . . 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 2848	. . . . . . . . 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 7486	. . . . . . . . 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 535	. . . . . . 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 7493	. . . . . . 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 1373	. . . . 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 2599	. . . 4 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) → ⊥)
58	14, 57	rexlimddv 2599	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → ⊥)
59	58	inegd 1372	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ 𝐵<P 𝑦)
60	59	ralrimiva 2550	1 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ 𝐵<P 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708   = wceq 1353  ⊥wfal 1358  ∃wex 1492   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  {crab 2459  Vcvv 2739   ⊆ wss 3131  ⟨cop 3597  ∪ cuni 3811  ∩ cint 3846   class class class wbr 4005  dom cdm 4628   “ cima 4631  Fun wfun 5212  ⟶wf 5214  –onto→wfo 5216  ‘cfv 5218  1^st c1st 6141  2^nd c2nd 6142  Qcnq 7281   <_Q cltq 7286  Pcnp 7292  <_P cltp 7296

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-2o 6420  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-enq0 7425  df-nq0 7426  df-0nq0 7427  df-plq0 7428  df-mq0 7429  df-inp 7467  df-iltp 7471

This theorem is referenced by:  suplocexpr  7726