Theorem noinfbnd1lem2 28093

Description: Lemma for noinfbnd1 28098. When there is no minimum, if any member of 𝐵 is a prolongment of 𝑇, then so are all elements below it. (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
noinfbnd1lem2	⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ ((𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑊))) → (𝑊 ↾ dom 𝑇) = 𝑇)

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

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

Proof of Theorem noinfbnd1lem2

Step	Hyp	Ref	Expression
1		simp3rl 1265	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ ((𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑊))) → 𝑊 ∈ 𝐵)
2		noinfbnd1.1	. . . 4 ⊢ 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))))
3	2	noinfbnd1lem1 28092	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑊 ∈ 𝐵) → ¬ (𝑊 ↾ dom 𝑇) <s 𝑇)
4	1, 3	syld3an3 1436	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ ((𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑊))) → ¬ (𝑊 ↾ dom 𝑇) <s 𝑇)
5		simp3rr 1266	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ ((𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑊))) → ¬ 𝑈 <s 𝑊)
6		simp2l 1218	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ ((𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑊))) → 𝐵 ⊆ No)
7		simp3ll 1263	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ ((𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑊))) → 𝑈 ∈ 𝐵)
8	6, 7	sseldd 3913	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ ((𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑊))) → 𝑈 ∈ No)
9	6, 1	sseldd 3913	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ ((𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑊))) → 𝑊 ∈ No)
10	2	noinfno 28087	. . . . . . 7 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) → 𝑇 ∈ No)
11	10	3ad2ant2 1152	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ ((𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑊))) → 𝑇 ∈ No)
12		nodmon 28019	. . . . . 6 ⊢ (𝑇 ∈ No → dom 𝑇 ∈ On)
13	11, 12	syl 18	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ ((𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑊))) → dom 𝑇 ∈ On)
14		ltsres 28031	. . . . 5 ⊢ ((𝑈 ∈ No ∧ 𝑊 ∈ No ∧ dom 𝑇 ∈ On) → ((𝑈 ↾ dom 𝑇) <s (𝑊 ↾ dom 𝑇) → 𝑈 <s 𝑊))
15	8, 9, 13, 14	syl3anc 1398	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ ((𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑊))) → ((𝑈 ↾ dom 𝑇) <s (𝑊 ↾ dom 𝑇) → 𝑈 <s 𝑊))
16	5, 15	mtod 201	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ ((𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑊))) → ¬ (𝑈 ↾ dom 𝑇) <s (𝑊 ↾ dom 𝑇))
17		simp3lr 1264	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ ((𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑊))) → (𝑈 ↾ dom 𝑇) = 𝑇)
18	17	breq1d 5115	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ ((𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑊))) → ((𝑈 ↾ dom 𝑇) <s (𝑊 ↾ dom 𝑇) ↔ 𝑇 <s (𝑊 ↾ dom 𝑇)))
19	16, 18	mtbid 327	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ ((𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑊))) → ¬ 𝑇 <s (𝑊 ↾ dom 𝑇))
20		noreson 28029	. . . 4 ⊢ ((𝑊 ∈ No ∧ dom 𝑇 ∈ On) → (𝑊 ↾ dom 𝑇) ∈ No)
21	9, 13, 20	syl2anc 596	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ ((𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑊))) → (𝑊 ↾ dom 𝑇) ∈ No)
22		ltsso 28045	. . . 4 ⊢ <s Or No
23		sotrieq2 5594	. . . 4 ⊢ (( <s Or No ∧ ((𝑊 ↾ dom 𝑇) ∈ No ∧ 𝑇 ∈ No)) → ((𝑊 ↾ dom 𝑇) = 𝑇 ↔ (¬ (𝑊 ↾ dom 𝑇) <s 𝑇 ∧ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇))))
24	22, 23	mpan 703	. . 3 ⊢ (((𝑊 ↾ dom 𝑇) ∈ No ∧ 𝑇 ∈ No) → ((𝑊 ↾ dom 𝑇) = 𝑇 ↔ (¬ (𝑊 ↾ dom 𝑇) <s 𝑇 ∧ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇))))
25	21, 11, 24	syl2anc 596	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ ((𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑊))) → ((𝑊 ↾ dom 𝑇) = 𝑇 ↔ (¬ (𝑊 ↾ dom 𝑇) <s 𝑇 ∧ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇))))
26	4, 19, 25	mpbir2and 726	1 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ ((𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑊))) → (𝑊 ↾ dom 𝑇) = 𝑇)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  {cab 2741  ∀wral 3079  ∃wrex 3089   ⊆ wss 3899   ∪ cun 4151  ifcif 4484  {csn 4587  ⟨cop 4837   class class class wbr 5105   ↦ cmpt 5188   Or wor 5561  dom cdm 5654   ↾ cres 5656  Oncon0 6368  suc csuc 6370  ℩cio 6498  ‘cfv 6546  ℩crio 7383  1_oc1o 8480  Nocsur 28009   <s clts 28010

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4838  df-uni 4870  df-int 4910  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6371  df-on 6372  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-fo 6552  df-fv 6554  df-riota 7384  df-1o 8487  df-2o 8488  df-no 28012  df-lts 28013  df-bday 28014

This theorem is used by:  noinfbnd1lem3  28094  noinfbnd1lem4  28095  noinfbnd1lem5  28096