Theorem noinfbnd1lem2 27694

Description: Lemma for noinfbnd1 27699. 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 1247	. . 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 27693	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑊 ∈ 𝐵) → ¬ (𝑊 ↾ dom 𝑇) <s 𝑇)
4	1, 3	syld3an3 1411	. 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 1248	. . . 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 1200	. . . . . 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 1245	. . . . . 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 3934	. . . . 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 3934	. . . . 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 27688	. . . . . . 7 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) → 𝑇 ∈ No )
11	10	3ad2ant2 1134	. . . . . 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 27620	. . . . . 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 27632	. . . . 5 ⊢ ((𝑈 ∈ No ∧ 𝑊 ∈ No ∧ dom 𝑇 ∈ On) → ((𝑈 ↾ dom 𝑇) <s (𝑊 ↾ dom 𝑇) → 𝑈 <s 𝑊))
15	8, 9, 13, 14	syl3anc 1373	. . . 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 1246	. . . 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 5108	. . 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 27630	. . . 4 ⊢ ((𝑊 ∈ No ∧ dom 𝑇 ∈ On) → (𝑊 ↾ dom 𝑇) ∈ No )
21	9, 13, 20	syl2anc 584	. . 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 27646	. . . 4 ⊢ <s Or No
23		sotrieq2 5564	. . . 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 690	. . 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 584	. 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 713	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 1086   = wceq 1541   ∈ wcel 2113  {cab 2714  ∀wral 3051  ∃wrex 3060   ∪ cun 3899   ⊆ wss 3901  ifcif 4479  {csn 4580  ⟨cop 4586   class class class wbr 5098   ↦ cmpt 5179   Or wor 5531  dom cdm 5624   ↾ cres 5626  Oncon0 6317  suc csuc 6319  ℩cio 6446  ‘cfv 6492  ℩crio 7314  1_oc1o 8390   No csur 27609   <s clts 27610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-riota 7315  df-1o 8397  df-2o 8398  df-no 27612  df-lts 27613  df-bday 27614

This theorem is referenced by:  noinfbnd1lem3  27695  noinfbnd1lem4  27696  noinfbnd1lem5  27697