Theorem nosupbnd1lem6 27766

Description: Lemma for nosupbnd1 27767. Establish a hard upper bound when there is no maximum. (Contributed by Scott Fenton, 6-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
nosupbnd1lem6	⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → (𝑈 ↾ dom 𝑆) <s 𝑆)

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

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

Proof of Theorem nosupbnd1lem6

Step	Hyp	Ref	Expression
1		simp2l 1196	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → 𝐴 ⊆ No )
2		simp3 1135	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → 𝑈 ∈ 𝐴)
3	1, 2	sseldd 3992	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → 𝑈 ∈ No )
4		nofv 27710	. . . . 5 ⊢ (𝑈 ∈ No → ((𝑈‘dom 𝑆) = ∅ ∨ (𝑈‘dom 𝑆) = 1o ∨ (𝑈‘dom 𝑆) = 2o))
5	3, 4	syl 17	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → ((𝑈‘dom 𝑆) = ∅ ∨ (𝑈‘dom 𝑆) = 1o ∨ (𝑈‘dom 𝑆) = 2o))
6		3oran 1106	. . . 4 ⊢ (((𝑈‘dom 𝑆) = ∅ ∨ (𝑈‘dom 𝑆) = 1o ∨ (𝑈‘dom 𝑆) = 2o) ↔ ¬ (¬ (𝑈‘dom 𝑆) = ∅ ∧ ¬ (𝑈‘dom 𝑆) = 1o ∧ ¬ (𝑈‘dom 𝑆) = 2o))
7	5, 6	sylib 217	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → ¬ (¬ (𝑈‘dom 𝑆) = ∅ ∧ ¬ (𝑈‘dom 𝑆) = 1o ∧ ¬ (𝑈‘dom 𝑆) = 2o))
8		simpl1 1188	. . . . . 6 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → ¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
9		simpl2 1189	. . . . . 6 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → (𝐴 ⊆ No ∧ 𝐴 ∈ V))
10		simpl3 1190	. . . . . 6 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → 𝑈 ∈ 𝐴)
11		simpr 483	. . . . . 6 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → (𝑈 ↾ dom 𝑆) = 𝑆)
12		nosupbnd1.1	. . . . . . 7 ⊢ 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))))
13	12	nosupbnd1lem4 27764	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ ∅)
14	8, 9, 10, 11, 13	syl112anc 1371	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → (𝑈‘dom 𝑆) ≠ ∅)
15	14	neneqd 2938	. . . 4 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → ¬ (𝑈‘dom 𝑆) = ∅)
16	12	nosupbnd1lem5 27765	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ 1o)
17	8, 9, 10, 11, 16	syl112anc 1371	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → (𝑈‘dom 𝑆) ≠ 1o)
18	17	neneqd 2938	. . . 4 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → ¬ (𝑈‘dom 𝑆) = 1o)
19	12	nosupbnd1lem3 27763	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ 2o)
20	8, 9, 10, 11, 19	syl112anc 1371	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → (𝑈‘dom 𝑆) ≠ 2o)
21	20	neneqd 2938	. . . 4 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → ¬ (𝑈‘dom 𝑆) = 2o)
22	15, 18, 21	3jca 1125	. . 3 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → (¬ (𝑈‘dom 𝑆) = ∅ ∧ ¬ (𝑈‘dom 𝑆) = 1o ∧ ¬ (𝑈‘dom 𝑆) = 2o))
23	7, 22	mtand 814	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → ¬ (𝑈 ↾ dom 𝑆) = 𝑆)
24	12	nosupbnd1lem1 27761	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → ¬ 𝑆 <s (𝑈 ↾ dom 𝑆))
25	12	nosupno 27756	. . . . . 6 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
26	25	3ad2ant2 1131	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → 𝑆 ∈ No )
27		nodmon 27703	. . . . 5 ⊢ (𝑆 ∈ No → dom 𝑆 ∈ On)
28	26, 27	syl 17	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → dom 𝑆 ∈ On)
29		noreson 27713	. . . 4 ⊢ ((𝑈 ∈ No ∧ dom 𝑆 ∈ On) → (𝑈 ↾ dom 𝑆) ∈ No )
30	3, 28, 29	syl2anc 582	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → (𝑈 ↾ dom 𝑆) ∈ No )
31		sltso 27729	. . . 4 ⊢ <s Or No
32		solin 5623	. . . 4 ⊢ (( <s Or No ∧ ((𝑈 ↾ dom 𝑆) ∈ No ∧ 𝑆 ∈ No )) → ((𝑈 ↾ dom 𝑆) <s 𝑆 ∨ (𝑈 ↾ dom 𝑆) = 𝑆 ∨ 𝑆 <s (𝑈 ↾ dom 𝑆)))
33	31, 32	mpan 688	. . 3 ⊢ (((𝑈 ↾ dom 𝑆) ∈ No ∧ 𝑆 ∈ No ) → ((𝑈 ↾ dom 𝑆) <s 𝑆 ∨ (𝑈 ↾ dom 𝑆) = 𝑆 ∨ 𝑆 <s (𝑈 ↾ dom 𝑆)))
34	30, 26, 33	syl2anc 582	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → ((𝑈 ↾ dom 𝑆) <s 𝑆 ∨ (𝑈 ↾ dom 𝑆) = 𝑆 ∨ 𝑆 <s (𝑈 ↾ dom 𝑆)))
35	23, 24, 34	ecase23d 1470	1 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → (𝑈 ↾ dom 𝑆) <s 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   ∨ w3o 1083   ∧ w3a 1084   = wceq 1534   ∈ wcel 2100  {cab 2706   ≠ wne 2933  ∀wral 3054  ∃wrex 3063  Vcvv 3472   ∪ cun 3957   ⊆ wss 3959  ∅c0 4335  ifcif 4536  {csn 4636  ⟨cop 4642   class class class wbr 5156   ↦ cmpt 5239   Or wor 5597  dom cdm 5686   ↾ cres 5688  Oncon0 6380  suc csuc 6382  ℩cio 6508  ‘cfv 6558  ℩crio 7385  1_oc1o 8497  2_oc2o 8498   No csur 27692   <s cslt 27693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-rep 5293  ax-sep 5307  ax-nul 5314  ax-pow 5373  ax-pr 5437  ax-un 7751

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ne 2934  df-ral 3055  df-rex 3064  df-rmo 3373  df-reu 3374  df-rab 3429  df-v 3474  df-sbc 3789  df-csb 3905  df-dif 3962  df-un 3964  df-in 3966  df-ss 3976  df-pss 3979  df-nul 4336  df-if 4537  df-pw 4612  df-sn 4637  df-pr 4639  df-tp 4641  df-op 4643  df-uni 4919  df-int 4960  df-br 5157  df-opab 5219  df-mpt 5240  df-tr 5274  df-id 5584  df-eprel 5590  df-po 5598  df-so 5599  df-fr 5641  df-we 5643  df-xp 5692  df-rel 5693  df-cnv 5694  df-co 5695  df-dm 5696  df-rn 5697  df-res 5698  df-ima 5699  df-ord 6383  df-on 6384  df-suc 6386  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-fo 6564  df-fv 6566  df-riota 7386  df-1o 8504  df-2o 8505  df-no 27695  df-slt 27696  df-bday 27697

This theorem is referenced by:  nosupbnd1  27767