Theorem noetalem5 33221

Description: Lemma for noeta 33222. The full statement of the theorem with hypotheses. (Contributed by Scott Fenton, 7-Dec-2021.)

Hypotheses

Ref	Expression
noetalem.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))))
noetalem.2	⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}))

Assertion

Ref	Expression
noetalem5	⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → ∃𝑧 ∈ No (∀𝑎 ∈ 𝐴 𝑎 <s 𝑧 ∧ ∀𝑏 ∈ 𝐵 𝑧 <s 𝑏 ∧ ( bday ‘𝑧) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))

Distinct variable groups:   𝐴,𝑎,𝑏,𝑔   𝑢,𝑎,𝐴,𝑣,𝑥,𝑦   𝑧,𝑎,𝐴   𝐵,𝑎,𝑏   𝑔,𝑏,𝑥   𝑧,𝑏,𝐵   𝑢,𝑔,𝑣,𝑥,𝑦   𝑆,𝑎,𝑔   𝑣,𝑢,𝑥,𝑦   𝑍,𝑎,𝑏,𝑧

Allowed substitution hints:   𝐵(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑆(𝑥,𝑦,𝑧,𝑣,𝑢,𝑏)   𝑉(𝑥,𝑦,𝑧,𝑣,𝑢,𝑔,𝑎,𝑏)   𝑊(𝑥,𝑦,𝑧,𝑣,𝑢,𝑔,𝑎,𝑏)   𝑍(𝑥,𝑦,𝑣,𝑢,𝑔)

Proof of Theorem noetalem5

Step	Hyp	Ref	Expression
1		elex 3512	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2	1	anim2i 618	. 2 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) → (𝐴 ⊆ No ∧ 𝐴 ∈ V))
3		elex 3512	. . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
4	3	anim2i 618	. 2 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) → (𝐵 ⊆ No ∧ 𝐵 ∈ V))
5		id 22	. 2 ⊢ (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏)
6		simp1l 1193	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → 𝐴 ⊆ No )
7		simp1r 1194	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → 𝐴 ∈ V)
8		simp2r 1196	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → 𝐵 ∈ V)
9		noetalem.1	. . . . 5 ⊢ 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))))
10		noetalem.2	. . . . 5 ⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}))
11	9, 10	noetalem1 33217	. . . 4 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑍 ∈ No )
12	6, 7, 8, 11	syl3anc 1367	. . 3 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → 𝑍 ∈ No )
13		simplll 773	. . . . . 6 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ 𝑎 ∈ 𝐴) → 𝐴 ⊆ No )
14		simpllr 774	. . . . . 6 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ 𝑎 ∈ 𝐴) → 𝐴 ∈ V)
15		simplrr 776	. . . . . 6 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ 𝑎 ∈ 𝐴) → 𝐵 ∈ V)
16		simpr 487	. . . . . 6 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
17	9, 10	noetalem2 33218	. . . . . 6 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑎 ∈ 𝐴) → 𝑎 <s 𝑍)
18	13, 14, 15, 16, 17	syl31anc 1369	. . . . 5 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ 𝑎 ∈ 𝐴) → 𝑎 <s 𝑍)
19	18	ralrimiva 3182	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)
20	19	3adant3 1128	. . 3 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)
21	9, 10	noetalem3 33219	. . 3 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏)
22	9, 10	noetalem4 33220	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → ( bday ‘𝑍) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
23	22	3adant3 1128	. . 3 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → ( bday ‘𝑍) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))
24		breq2 5070	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑎 <s 𝑧 ↔ 𝑎 <s 𝑍))
25	24	ralbidv 3197	. . . . 5 ⊢ (𝑧 = 𝑍 → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑧 ↔ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍))
26		breq1 5069	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑧 <s 𝑏 ↔ 𝑍 <s 𝑏))
27	26	ralbidv 3197	. . . . 5 ⊢ (𝑧 = 𝑍 → (∀𝑏 ∈ 𝐵 𝑧 <s 𝑏 ↔ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏))
28		fveq2 6670	. . . . . 6 ⊢ (𝑧 = 𝑍 → ( bday ‘𝑧) = ( bday ‘𝑍))
29	28	sseq1d 3998	. . . . 5 ⊢ (𝑧 = 𝑍 → (( bday ‘𝑧) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)) ↔ ( bday ‘𝑍) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))
30	25, 27, 29	3anbi123d 1432	. . . 4 ⊢ (𝑧 = 𝑍 → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑧 ∧ ∀𝑏 ∈ 𝐵 𝑧 <s 𝑏 ∧ ( bday ‘𝑧) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))) ↔ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑍 ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏 ∧ ( bday ‘𝑍) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))))
31	30	rspcev 3623	. . 3 ⊢ ((𝑍 ∈ No ∧ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑍 ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏 ∧ ( bday ‘𝑍) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))) → ∃𝑧 ∈ No (∀𝑎 ∈ 𝐴 𝑎 <s 𝑧 ∧ ∀𝑏 ∈ 𝐵 𝑧 <s 𝑏 ∧ ( bday ‘𝑧) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))
32	12, 20, 21, 23, 31	syl13anc 1368	. 2 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → ∃𝑧 ∈ No (∀𝑎 ∈ 𝐴 𝑎 <s 𝑧 ∧ ∀𝑏 ∈ 𝐵 𝑧 <s 𝑏 ∧ ( bday ‘𝑧) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))
33	2, 4, 5, 32	syl3an 1156	1 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → ∃𝑧 ∈ No (∀𝑎 ∈ 𝐴 𝑎 <s 𝑧 ∧ ∀𝑏 ∈ 𝐵 𝑧 <s 𝑏 ∧ ( bday ‘𝑧) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  {cab 2799  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ∖ cdif 3933   ∪ cun 3934   ⊆ wss 3936  ifcif 4467  {csn 4567  ⟨cop 4573  ∪ cuni 4838   class class class wbr 5066   ↦ cmpt 5146   × cxp 5553  dom cdm 5555   ↾ cres 5557   “ cima 5558  suc csuc 6193  ℩cio 6312  ‘cfv 6355  ℩crio 7113  1_oc1o 8095  2_oc2o 8096   No csur 33147   <s cslt 33148   bday cbday 33149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-ord 6194  df-on 6195  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-1o 8102  df-2o 8103  df-no 33150  df-slt 33151  df-bday 33152

This theorem is referenced by:  noeta  33222