Theorem nosupbnd1lem6 27659

Description: Lemma for nosupbnd1 27660. 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 3974	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → 𝑈 ∈ No )
4		nofv 27603	. . . . 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 27657	. . . . . 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 2935	. . . 4 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → ¬ (𝑈‘dom 𝑆) = ∅)
16	12	nosupbnd1lem5 27658	. . . . . 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 2935	. . . 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 27656	. . . . . 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 2935	. . . 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 27654	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → ¬ 𝑆 <s (𝑈 ↾ dom 𝑆))
25	12	nosupno 27649	. . . . . 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 27596	. . . . 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 27606	. . . 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 27622	. . . 4 ⊢ <s Or No
32		solin 5610	. . . 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 1469	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 1533   ∈ wcel 2098  {cab 2702   ≠ wne 2930  ∀wral 3051  ∃wrex 3060  Vcvv 3463   ∪ cun 3939   ⊆ wss 3941  ∅c0 4319  ifcif 4525  {csn 4625  ⟨cop 4631   class class class wbr 5144   ↦ cmpt 5227   Or wor 5584  dom cdm 5673   ↾ cres 5675  Oncon0 6365  suc csuc 6367  ℩cio 6493  ‘cfv 6543  ℩crio 7368  1_oc1o 8473  2_oc2o 8474   No csur 27586   <s cslt 27587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pr 5424  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-1o 8480  df-2o 8481  df-no 27589  df-slt 27590  df-bday 27591

This theorem is referenced by:  nosupbnd1  27660