Theorem noinfbnd1lem6 27860

Description: Lemma for noinfbnd1 27861. 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   𝑔,𝑉   𝑥,𝑈,𝑦

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 1216	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → 𝐵 ⊆ No )
2		simp3 1154	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → 𝑈 ∈ 𝐵)
3	1, 2	sseldd 3946	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → 𝑈 ∈ No )
4		nofv 27789	. . . . 5 ⊢ (𝑈 ∈ No → ((𝑈‘dom 𝑇) = ∅ ∨ (𝑈‘dom 𝑇) = 1o ∨ (𝑈‘dom 𝑇) = 2o))
5	3, 4	syl 18	. . . 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 1124	. . . 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 221	. . 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 1208	. . . . . 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 1209	. . . . . 6 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) ∧ 𝑇 = (𝑈 ↾ dom 𝑇)) → (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉))
10		simpl3 1210	. . . . . 6 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) ∧ 𝑇 = (𝑈 ↾ dom 𝑇)) → 𝑈 ∈ 𝐵)
11		simpr 489	. . . . . . 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 2775	. . . . . 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 27858	. . . . . 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 1399	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) ∧ 𝑇 = (𝑈 ↾ dom 𝑇)) → (𝑈‘dom 𝑇) ≠ ∅)
16	15	neneqd 2969	. . . 4 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) ∧ 𝑇 = (𝑈 ↾ dom 𝑇)) → ¬ (𝑈‘dom 𝑇) = ∅)
17	13	noinfbnd1lem3 27857	. . . . . 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 1399	. . . . 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 2969	. . . 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 27859	. . . . . 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 1399	. . . . 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 2969	. . . 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 1144	. . 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 827	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → ¬ 𝑇 = (𝑈 ↾ dom 𝑇))
25	13	noinfbnd1lem1 27855	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → ¬ (𝑈 ↾ dom 𝑇) <s 𝑇)
26	13	noinfno 27850	. . . 4 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) → 𝑇 ∈ No )
27	26	3ad2ant2 1150	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → 𝑇 ∈ No )
28		nodmon 27782	. . . . 5 ⊢ (𝑇 ∈ No → dom 𝑇 ∈ On)
29	27, 28	syl 18	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → dom 𝑇 ∈ On)
30		noreson 27792	. . . 4 ⊢ ((𝑈 ∈ No ∧ dom 𝑇 ∈ On) → (𝑈 ↾ dom 𝑇) ∈ No )
31	3, 29, 30	syl2anc 595	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → (𝑈 ↾ dom 𝑇) ∈ No )
32		ltsso 27808	. . . 4 ⊢ <s Or No
33		solin 5599	. . . 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 702	. . 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 595	. 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 1500	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 400   ∨ w3o 1100   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  {cab 2747   ≠ wne 2964  ∀wral 3085  ∃wrex 3095   ∪ cun 3911   ⊆ wss 3913  ∅c0 4294  ifcif 4492  {csn 4594  ⟨cop 4600   class class class wbr 5113   ↦ cmpt 5196   Or wor 5571  dom cdm 5664   ↾ cres 5666  Oncon0 6363  suc csuc 6365  ℩cio 6493  ‘cfv 6539  ℩crio 7369  1_oc1o 8448  2_oc2o 8449   No csur 27772   <s clts 27773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5273  ax-pow 5339  ax-pr 5407  ax-un 7735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-int 4917  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5559  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-fo 6545  df-fv 6547  df-riota 7370  df-1o 8455  df-2o 8456  df-no 27775  df-lts 27776  df-bday 27777

This theorem is referenced by:  noinfbnd1  27861