Theorem noinfbnd1lem6 27697

Description: Lemma for noinfbnd1 27698. 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 1200	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → 𝐵 ⊆ No )
2		simp3 1138	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → 𝑈 ∈ 𝐵)
3	1, 2	sseldd 3964	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → 𝑈 ∈ No )
4		nofv 27626	. . . . 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 1108	. . . 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 218	. . 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 1192	. . . . . 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 1193	. . . . . 6 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) ∧ 𝑇 = (𝑈 ↾ dom 𝑇)) → (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉))
10		simpl3 1194	. . . . . 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 2742	. . . . . 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 27695	. . . . . 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 1376	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) ∧ 𝑇 = (𝑈 ↾ dom 𝑇)) → (𝑈‘dom 𝑇) ≠ ∅)
16	15	neneqd 2938	. . . 4 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) ∧ 𝑇 = (𝑈 ↾ dom 𝑇)) → ¬ (𝑈‘dom 𝑇) = ∅)
17	13	noinfbnd1lem3 27694	. . . . . 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 1376	. . . . 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 2938	. . . 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 27696	. . . . . 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 1376	. . . . 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 2938	. . . 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 1128	. . 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 27692	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → ¬ (𝑈 ↾ dom 𝑇) <s 𝑇)
26	13	noinfno 27687	. . . 4 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) → 𝑇 ∈ No )
27	26	3ad2ant2 1134	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → 𝑇 ∈ No )
28		nodmon 27619	. . . . 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 27629	. . . 4 ⊢ ((𝑈 ∈ No ∧ dom 𝑇 ∈ On) → (𝑈 ↾ dom 𝑇) ∈ No )
31	3, 29, 30	syl2anc 584	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → (𝑈 ↾ dom 𝑇) ∈ No )
32		sltso 27645	. . . 4 ⊢ <s Or No
33		solin 5593	. . . 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 690	. . 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 584	. 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 1475	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 1085   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {cab 2714   ≠ wne 2933  ∀wral 3052  ∃wrex 3061   ∪ cun 3929   ⊆ wss 3931  ∅c0 4313  ifcif 4505  {csn 4606  ⟨cop 4612   class class class wbr 5124   ↦ cmpt 5206   Or wor 5565  dom cdm 5659   ↾ cres 5661  Oncon0 6357  suc csuc 6359  ℩cio 6487  ‘cfv 6536  ℩crio 7366  1_oc1o 8478  2_oc2o 8479   No csur 27608   <s cslt 27609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-int 4928  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-riota 7367  df-1o 8485  df-2o 8486  df-no 27611  df-slt 27612  df-bday 27613

This theorem is referenced by:  noinfbnd1  27698