Theorem nosupbnd1lem2 27638

Description: Lemma for nosupbnd1 27643. 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 1248	. . . 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 1200	. . . . . 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 1247	. . . . . 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 3938	. . . . 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 1245	. . . . . 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 3938	. . . . 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 27632	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
9	8	3ad2ant2 1134	. . . . . 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 27579	. . . . . 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 27591	. . . . 5 ⊢ ((𝑊 ∈ No ∧ 𝑈 ∈ No ∧ dom 𝑆 ∈ On) → ((𝑊 ↾ dom 𝑆) <s (𝑈 ↾ dom 𝑆) → 𝑊 <s 𝑈))
13	4, 6, 11, 12	syl3anc 1373	. . . 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 198	. . 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 1246	. . . 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 5107	. . 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 324	. 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 27637	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑊 ∈ 𝐴) → ¬ 𝑆 <s (𝑊 ↾ dom 𝑆))
19	3, 18	syld3an3 1411	. 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 27589	. . . 4 ⊢ ((𝑊 ∈ No ∧ dom 𝑆 ∈ On) → (𝑊 ↾ dom 𝑆) ∈ No )
21	4, 11, 20	syl2anc 584	. . 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 27605	. . . 4 ⊢ <s Or No
23		sotrieq2 5563	. . . 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 690	. . 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 584	. 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 713	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053  Vcvv 3438   ∪ cun 3903   ⊆ wss 3905  ifcif 4478  {csn 4579  ⟨cop 4585   class class class wbr 5095   ↦ cmpt 5176   Or wor 5530  dom cdm 5623   ↾ cres 5625  Oncon0 6311  suc csuc 6313  ℩cio 6440  ‘cfv 6486  ℩crio 7309  2_oc2o 8389   No csur 27568   <s cslt 27569

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 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4862  df-int 4900  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ord 6314  df-on 6315  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fo 6492  df-fv 6494  df-riota 7310  df-1o 8395  df-2o 8396  df-no 27571  df-slt 27572  df-bday 27573

This theorem is referenced by:  nosupbnd1lem3  27639  nosupbnd1lem4  27640  nosupbnd1lem5  27641