Theorem noinfbnd1lem6 27648

Description: Lemma for noinfbnd1 27649. Establish a hard lower bound when there is no minimum. (Contributed by Scott Fenton, 9-Aug-2024.)

Hypothesis

Ref	Expression
noinfbnd1.1	⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))

Assertion

Ref	Expression
noinfbnd1lem6	⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → 𝑇 <s (𝑈 ↾ dom 𝑇))

Distinct variable groups:   𝐵,𝑔,𝑢,𝑣,𝑥,𝑦   𝑣,𝑈   𝑥,𝑢,𝑦   𝑔,𝑉   𝑥,𝑣,𝑦,𝑈

Allowed substitution hints:   𝑇(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑈(𝑢,𝑔)   𝑉(𝑥,𝑦,𝑣,𝑢)

Proof of Theorem noinfbnd1lem6

Step	Hyp	Ref	Expression
1		simp2l 1197	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → 𝐵 ⊆ No )
2		simp3 1136	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → 𝑈 ∈ 𝐵)
3	1, 2	sseldd 3979	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → 𝑈 ∈ No )
4		nofv 27577	. . . . 5 ⊢ (𝑈 ∈ No → ((𝑈‘dom 𝑇) = ∅ ∨ (𝑈‘dom 𝑇) = 1o ∨ (𝑈‘dom 𝑇) = 2o))
5	3, 4	syl 17	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → ((𝑈‘dom 𝑇) = ∅ ∨ (𝑈‘dom 𝑇) = 1o ∨ (𝑈‘dom 𝑇) = 2o))
6		3oran 1107	. . . 4 ⊢ (((𝑈‘dom 𝑇) = ∅ ∨ (𝑈‘dom 𝑇) = 1o ∨ (𝑈‘dom 𝑇) = 2o) ↔ ¬ (¬ (𝑈‘dom 𝑇) = ∅ ∧ ¬ (𝑈‘dom 𝑇) = 1o ∧ ¬ (𝑈‘dom 𝑇) = 2o))
7	5, 6	sylib 217	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → ¬ (¬ (𝑈‘dom 𝑇) = ∅ ∧ ¬ (𝑈‘dom 𝑇) = 1o ∧ ¬ (𝑈‘dom 𝑇) = 2o))
8		simpl1 1189	. . . . . 6 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) ∧ 𝑇 = (𝑈 ↾ dom 𝑇)) → ¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥)
9		simpl2 1190	. . . . . 6 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) ∧ 𝑇 = (𝑈 ↾ dom 𝑇)) → (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉))
10		simpl3 1191	. . . . . 6 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) ∧ 𝑇 = (𝑈 ↾ dom 𝑇)) → 𝑈 ∈ 𝐵)
11		simpr 484	. . . . . . 7 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) ∧ 𝑇 = (𝑈 ↾ dom 𝑇)) → 𝑇 = (𝑈 ↾ dom 𝑇))
12	11	eqcomd 2733	. . . . . 6 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) ∧ 𝑇 = (𝑈 ↾ dom 𝑇)) → (𝑈 ↾ dom 𝑇) = 𝑇)
13		noinfbnd1.1	. . . . . . 7 ⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
14	13	noinfbnd1lem4 27646	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → (𝑈‘dom 𝑇) ≠ ∅)
15	8, 9, 10, 12, 14	syl112anc 1372	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) ∧ 𝑇 = (𝑈 ↾ dom 𝑇)) → (𝑈‘dom 𝑇) ≠ ∅)
16	15	neneqd 2940	. . . 4 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) ∧ 𝑇 = (𝑈 ↾ dom 𝑇)) → ¬ (𝑈‘dom 𝑇) = ∅)
17	13	noinfbnd1lem3 27645	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → (𝑈‘dom 𝑇) ≠ 1o)
18	8, 9, 10, 12, 17	syl112anc 1372	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) ∧ 𝑇 = (𝑈 ↾ dom 𝑇)) → (𝑈‘dom 𝑇) ≠ 1o)
19	18	neneqd 2940	. . . 4 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) ∧ 𝑇 = (𝑈 ↾ dom 𝑇)) → ¬ (𝑈‘dom 𝑇) = 1o)
20	13	noinfbnd1lem5 27647	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → (𝑈‘dom 𝑇) ≠ 2o)
21	8, 9, 10, 12, 20	syl112anc 1372	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) ∧ 𝑇 = (𝑈 ↾ dom 𝑇)) → (𝑈‘dom 𝑇) ≠ 2o)
22	21	neneqd 2940	. . . 4 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) ∧ 𝑇 = (𝑈 ↾ dom 𝑇)) → ¬ (𝑈‘dom 𝑇) = 2o)
23	16, 19, 22	3jca 1126	. . 3 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) ∧ 𝑇 = (𝑈 ↾ dom 𝑇)) → (¬ (𝑈‘dom 𝑇) = ∅ ∧ ¬ (𝑈‘dom 𝑇) = 1o ∧ ¬ (𝑈‘dom 𝑇) = 2o))
24	7, 23	mtand 815	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → ¬ 𝑇 = (𝑈 ↾ dom 𝑇))
25	13	noinfbnd1lem1 27643	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → ¬ (𝑈 ↾ dom 𝑇) <s 𝑇)
26	13	noinfno 27638	. . . 4 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) → 𝑇 ∈ No )
27	26	3ad2ant2 1132	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → 𝑇 ∈ No )
28		nodmon 27570	. . . . 5 ⊢ (𝑇 ∈ No → dom 𝑇 ∈ On)
29	27, 28	syl 17	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → dom 𝑇 ∈ On)
30		noreson 27580	. . . 4 ⊢ ((𝑈 ∈ No ∧ dom 𝑇 ∈ On) → (𝑈 ↾ dom 𝑇) ∈ No )
31	3, 29, 30	syl2anc 583	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → (𝑈 ↾ dom 𝑇) ∈ No )
32		sltso 27596	. . . 4 ⊢ <s Or No
33		solin 5609	. . . 4 ⊢ (( <s Or No ∧ (𝑇 ∈ No ∧ (𝑈 ↾ dom 𝑇) ∈ No )) → (𝑇 <s (𝑈 ↾ dom 𝑇) ∨ 𝑇 = (𝑈 ↾ dom 𝑇) ∨ (𝑈 ↾ dom 𝑇) <s 𝑇))
34	32, 33	mpan 689	. . 3 ⊢ ((𝑇 ∈ No ∧ (𝑈 ↾ dom 𝑇) ∈ No ) → (𝑇 <s (𝑈 ↾ dom 𝑇) ∨ 𝑇 = (𝑈 ↾ dom 𝑇) ∨ (𝑈 ↾ dom 𝑇) <s 𝑇))
35	27, 31, 34	syl2anc 583	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → (𝑇 <s (𝑈 ↾ dom 𝑇) ∨ 𝑇 = (𝑈 ↾ dom 𝑇) ∨ (𝑈 ↾ dom 𝑇) <s 𝑇))
36	24, 25, 35	ecase23d 1470	1 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → 𝑇 <s (𝑈 ↾ dom 𝑇))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ w3o 1084   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  {cab 2704   ≠ wne 2935  ∀wral 3056  ∃wrex 3065   ∪ cun 3942   ⊆ wss 3944  ∅c0 4318  ifcif 4524  {csn 4624  ⟨cop 4630   class class class wbr 5142   ↦ cmpt 5225   Or wor 5583  dom cdm 5672   ↾ cres 5674  Oncon0 6363  suc csuc 6365  ℩cio 6492  ‘cfv 6542  ℩crio 7369  1_oc1o 8473  2_oc2o 8474   No csur 27560   <s cslt 27561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-1o 8480  df-2o 8481  df-no 27563  df-slt 27564  df-bday 27565

This theorem is referenced by:  noinfbnd1  27649