Theorem noinfbnd1lem2 27688

Description: Lemma for noinfbnd1 27693. 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 1248	. . 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 27687	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑊 ∈ 𝐵) → ¬ (𝑊 ↾ dom 𝑇) <s 𝑇)
4	1, 3	syld3an3 1412	. 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 1249	. . . 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 1201	. . . . . 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 1246	. . . . . 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 3922	. . . . 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 3922	. . . . 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 27682	. . . . . . 7 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) → 𝑇 ∈ No )
11	10	3ad2ant2 1135	. . . . . 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 27614	. . . . . 6 ⊢ (𝑇 ∈ No → dom 𝑇 ∈ On)
13	11, 12	syl 17	. . . . 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 27626	. . . . 5 ⊢ ((𝑈 ∈ No ∧ 𝑊 ∈ No ∧ dom 𝑇 ∈ On) → ((𝑈 ↾ dom 𝑇) <s (𝑊 ↾ dom 𝑇) → 𝑈 <s 𝑊))
15	8, 9, 13, 14	syl3anc 1374	. . . 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 198	. . 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 1247	. . . 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 5095	. . 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 324	. 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 27624	. . . 4 ⊢ ((𝑊 ∈ No ∧ dom 𝑇 ∈ On) → (𝑊 ↾ dom 𝑇) ∈ No )
21	9, 13, 20	syl2anc 585	. . 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 27640	. . . 4 ⊢ <s Or No
23		sotrieq2 5571	. . . 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 691	. . 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 585	. 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 714	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)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {cab 2714  ∀wral 3051  ∃wrex 3061   ∪ cun 3887   ⊆ wss 3889  ifcif 4466  {csn 4567  ⟨cop 4573   class class class wbr 5085   ↦ cmpt 5166   Or wor 5538  dom cdm 5631   ↾ cres 5633  Oncon0 6323  suc csuc 6325  ℩cio 6452  ‘cfv 6498  ℩crio 7323  1_oc1o 8398   No csur 27603   <s clts 27604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fo 6504  df-fv 6506  df-riota 7324  df-1o 8405  df-2o 8406  df-no 27606  df-lts 27607  df-bday 27608

This theorem is referenced by:  noinfbnd1lem3  27689  noinfbnd1lem4  27690  noinfbnd1lem5  27691