Theorem nosupbnd1lem6 33843

Description: Lemma for nosupbnd1 33844. 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 1197	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → 𝐴 ⊆ No )
2		simp3 1136	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → 𝑈 ∈ 𝐴)
3	1, 2	sseldd 3918	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → 𝑈 ∈ No )
4		nofv 33787	. . . . 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 1107	. . . 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 1189	. . . . . 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 1190	. . . . . 6 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → (𝐴 ⊆ No ∧ 𝐴 ∈ V))
10		simpl3 1191	. . . . . 6 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → 𝑈 ∈ 𝐴)
11		simpr 484	. . . . . 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 33841	. . . . . 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 1372	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → (𝑈‘dom 𝑆) ≠ ∅)
15	14	neneqd 2947	. . . 4 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → ¬ (𝑈‘dom 𝑆) = ∅)
16	12	nosupbnd1lem5 33842	. . . . . 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 1372	. . . . 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 2947	. . . 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 33840	. . . . . 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 1372	. . . . 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 2947	. . . 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 1126	. . 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 812	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → ¬ (𝑈 ↾ dom 𝑆) = 𝑆)
24	12	nosupbnd1lem1 33838	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → ¬ 𝑆 <s (𝑈 ↾ dom 𝑆))
25	12	nosupno 33833	. . . . . 6 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
26	25	3ad2ant2 1132	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → 𝑆 ∈ No )
27		nodmon 33780	. . . . 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 33790	. . . 4 ⊢ ((𝑈 ∈ No ∧ dom 𝑆 ∈ On) → (𝑈 ↾ dom 𝑆) ∈ No )
30	3, 28, 29	syl2anc 583	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → (𝑈 ↾ dom 𝑆) ∈ No )
31		sltso 33806	. . . 4 ⊢ <s Or No
32		solin 5519	. . . 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 686	. . 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 583	. 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 1471	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 395   ∨ w3o 1084   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  {cab 2715   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∪ cun 3881   ⊆ wss 3883  ∅c0 4253  ifcif 4456  {csn 4558  ⟨cop 4564   class class class wbr 5070   ↦ cmpt 5153   Or wor 5493  dom cdm 5580   ↾ cres 5582  Oncon0 6251  suc csuc 6253  ℩cio 6374  ‘cfv 6418  ℩crio 7211  1_oc1o 8260  2_oc2o 8261   No csur 33770   <s cslt 33771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-1o 8267  df-2o 8268  df-no 33773  df-slt 33774  df-bday 33775

This theorem is referenced by:  nosupbnd1  33844