Theorem nosupbnd1lem2 32736

Description: Lemma for nosupbnd1 32741. When there is no maximum, if any member of 𝐴 is a prolongment of 𝑆, then so are all elements of 𝐴 above it. (Contributed by Scott Fenton, 5-Dec-2021.)

Hypothesis

Ref	Expression
nosupbnd1.1	⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))

Assertion

Ref	Expression
nosupbnd1lem2	⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ ((𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊 ∈ 𝐴 ∧ ¬ 𝑊 <s 𝑈))) → (𝑊 ↾ dom 𝑆) = 𝑆)

Distinct variable groups:   𝐴,𝑔,𝑢,𝑣,𝑥,𝑦   𝑣,𝑊   𝑥,𝑦

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

Proof of Theorem nosupbnd1lem2

Step	Hyp	Ref	Expression
1		simp3rr 1227	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ ((𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊 ∈ 𝐴 ∧ ¬ 𝑊 <s 𝑈))) → ¬ 𝑊 <s 𝑈)
2		simp2l 1179	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ ((𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊 ∈ 𝐴 ∧ ¬ 𝑊 <s 𝑈))) → 𝐴 ⊆ No )
3		simp3rl 1226	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ ((𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊 ∈ 𝐴 ∧ ¬ 𝑊 <s 𝑈))) → 𝑊 ∈ 𝐴)
4	2, 3	sseldd 3859	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ ((𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊 ∈ 𝐴 ∧ ¬ 𝑊 <s 𝑈))) → 𝑊 ∈ No )
5		simp3ll 1224	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ ((𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊 ∈ 𝐴 ∧ ¬ 𝑊 <s 𝑈))) → 𝑈 ∈ 𝐴)
6	2, 5	sseldd 3859	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ ((𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊 ∈ 𝐴 ∧ ¬ 𝑊 <s 𝑈))) → 𝑈 ∈ No )
7		nosupbnd1.1	. . . . . . . 8 ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
8	7	nosupno 32730	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
9	8	3ad2ant2 1114	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ ((𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊 ∈ 𝐴 ∧ ¬ 𝑊 <s 𝑈))) → 𝑆 ∈ No )
10		nodmon 32684	. . . . . 6 ⊢ (𝑆 ∈ No → dom 𝑆 ∈ On)
11	9, 10	syl 17	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ ((𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊 ∈ 𝐴 ∧ ¬ 𝑊 <s 𝑈))) → dom 𝑆 ∈ On)
12		sltres 32696	. . . . 5 ⊢ ((𝑊 ∈ No ∧ 𝑈 ∈ No ∧ dom 𝑆 ∈ On) → ((𝑊 ↾ dom 𝑆) <s (𝑈 ↾ dom 𝑆) → 𝑊 <s 𝑈))
13	4, 6, 11, 12	syl3anc 1351	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ ((𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊 ∈ 𝐴 ∧ ¬ 𝑊 <s 𝑈))) → ((𝑊 ↾ dom 𝑆) <s (𝑈 ↾ dom 𝑆) → 𝑊 <s 𝑈))
14	1, 13	mtod 190	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ ((𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊 ∈ 𝐴 ∧ ¬ 𝑊 <s 𝑈))) → ¬ (𝑊 ↾ dom 𝑆) <s (𝑈 ↾ dom 𝑆))
15		simp3lr 1225	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ ((𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊 ∈ 𝐴 ∧ ¬ 𝑊 <s 𝑈))) → (𝑈 ↾ dom 𝑆) = 𝑆)
16	15	breq2d 4941	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ ((𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊 ∈ 𝐴 ∧ ¬ 𝑊 <s 𝑈))) → ((𝑊 ↾ dom 𝑆) <s (𝑈 ↾ dom 𝑆) ↔ (𝑊 ↾ dom 𝑆) <s 𝑆))
17	14, 16	mtbid 316	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ ((𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊 ∈ 𝐴 ∧ ¬ 𝑊 <s 𝑈))) → ¬ (𝑊 ↾ dom 𝑆) <s 𝑆)
18	7	nosupbnd1lem1 32735	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑊 ∈ 𝐴) → ¬ 𝑆 <s (𝑊 ↾ dom 𝑆))
19	3, 18	syld3an3 1389	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ ((𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊 ∈ 𝐴 ∧ ¬ 𝑊 <s 𝑈))) → ¬ 𝑆 <s (𝑊 ↾ dom 𝑆))
20		noreson 32694	. . . 4 ⊢ ((𝑊 ∈ No ∧ dom 𝑆 ∈ On) → (𝑊 ↾ dom 𝑆) ∈ No )
21	4, 11, 20	syl2anc 576	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ ((𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊 ∈ 𝐴 ∧ ¬ 𝑊 <s 𝑈))) → (𝑊 ↾ dom 𝑆) ∈ No )
22		sltso 32708	. . . 4 ⊢ <s Or No
23		sotrieq2 5355	. . . 4 ⊢ (( <s Or No ∧ ((𝑊 ↾ dom 𝑆) ∈ No ∧ 𝑆 ∈ No )) → ((𝑊 ↾ dom 𝑆) = 𝑆 ↔ (¬ (𝑊 ↾ dom 𝑆) <s 𝑆 ∧ ¬ 𝑆 <s (𝑊 ↾ dom 𝑆))))
24	22, 23	mpan 677	. . 3 ⊢ (((𝑊 ↾ dom 𝑆) ∈ No ∧ 𝑆 ∈ No ) → ((𝑊 ↾ dom 𝑆) = 𝑆 ↔ (¬ (𝑊 ↾ dom 𝑆) <s 𝑆 ∧ ¬ 𝑆 <s (𝑊 ↾ dom 𝑆))))
25	21, 9, 24	syl2anc 576	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ ((𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊 ∈ 𝐴 ∧ ¬ 𝑊 <s 𝑈))) → ((𝑊 ↾ dom 𝑆) = 𝑆 ↔ (¬ (𝑊 ↾ dom 𝑆) <s 𝑆 ∧ ¬ 𝑆 <s (𝑊 ↾ dom 𝑆))))
26	17, 19, 25	mpbir2and 700	1 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ ((𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊 ∈ 𝐴 ∧ ¬ 𝑊 <s 𝑈))) → (𝑊 ↾ dom 𝑆) = 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050  {cab 2758  ∀wral 3088  ∃wrex 3089  Vcvv 3415   ∪ cun 3827   ⊆ wss 3829  ifcif 4350  {csn 4441  ⟨cop 4447   class class class wbr 4929   ↦ cmpt 5008   Or wor 5325  dom cdm 5407   ↾ cres 5409  Oncon0 6029  suc csuc 6031  ℩cio 6150  ‘cfv 6188  ℩crio 6936  2_oc2o 7899   No csur 32674   <s cslt 32675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-ord 6032  df-on 6033  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-1o 7905  df-2o 7906  df-no 32677  df-slt 32678  df-bday 32679

This theorem is referenced by:  nosupbnd1lem3  32737  nosupbnd1lem4  32738  nosupbnd1lem5  32739