Theorem suplocexprlemub 8038

Description: Lemma for suplocexpr 8040. 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 8037	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ P)
7	6	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → 𝐵 ∈ P)
8	2, 3, 4	suplocexprlemss 8030	. . . . . . . 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 3239	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → 𝑦 ∈ P)
12		ltdfpr 7821	. . . . . 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 8029	. . . . . . . . . . 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 2302	. . . . . . . . 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 4113	. . . . . . . . 9 ⊢ (𝑢 = 𝑠 → (𝑤 <Q 𝑢 ↔ 𝑤 <Q 𝑠))
22	21	rexbidv 2543	. . . . . . . 8 ⊢ (𝑢 = 𝑠 → (∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑠))
23	22	elrab 2973	. . . . . . 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 7790	. . . . . . . . . 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 2296	. . . . . . . . . 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 2816	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
35	34	elint2 3956	. . . . . . . . . . 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 6352	. . . . . . . . . . . . 13 ⊢ 2nd :V–onto→V
38		fofun 5591	. . . . . . . . . . . . 13 ⊢ (2nd :V–onto→V → Fun 2nd )
39	37, 38	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 2nd
40		vex 2816	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
41		fof 5590	. . . . . . . . . . . . . . 15 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
42	37, 41	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 2nd :V⟶V
43	42	fdmi 5516	. . . . . . . . . . . . 13 ⊢ dom 2nd = V
44	40, 43	eleqtrri 2308	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ dom 2nd
45		funfvima 5918	. . . . . . . . . . . 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 2926	. . . . . . . . 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 7800	. . . . . . . . 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 7807	. . . . . . 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 2665	. . . 4 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝐵) ∧ 𝑠 ∈ (1st ‘𝑦)))) → ⊥)
58	14, 57	rexlimddv 2665	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝐵<P 𝑦) → ⊥)
59	58	inegd 1417	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ 𝐵<P 𝑦)
60	59	ralrimiva 2615	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 2203  ∀wral 2520  ∃wrex 2521  {crab 2524  Vcvv 2813   ⊆ wss 3211  ⟨cop 3692  ∪ cuni 3914  ∩ cint 3949   class class class wbr 4109  dom cdm 4749   “ cima 4752  Fun wfun 5346  ⟶wf 5348  –onto→wfo 5350  ‘cfv 5352  1^st c1st 6332  2^nd c2nd 6333  Qcnq 7595   <_Q cltq 7600  Pcnp 7606  <_P cltp 7610

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-iltp 7785

This theorem is referenced by:  suplocexpr  8040