Theorem noetalem2 27224

Description: Lemma for noeta 27225. The full statement of the theorem with hypotheses in place. (Contributed by Scott Fenton, 10-Aug-2024.)

Hypotheses

Ref	Expression
noetalem2.1	⊢ 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))))
noetalem2.2	⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))

Assertion

Ref	Expression
noetalem2	⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ∃𝑐 ∈ No (∀𝑎 ∈ 𝐴 𝑎 <s 𝑐 ∧ ∀𝑏 ∈ 𝐵 𝑐 <s 𝑏 ∧ ( bday ‘𝑐) ⊆ 𝑂))

Distinct variable groups:   𝐴,𝑎,𝑏,𝑐   𝑔,𝑎,𝐴,𝑢,𝑣,𝑥,𝑦   𝐵,𝑎,𝑏,𝑐   𝑔,𝑏,𝐵   𝑢,𝐵   𝑣,𝑏,𝐵,𝑥,𝑦   𝑢,𝑔,𝑣,𝑥,𝑦   𝑂,𝑐   𝑢,𝑂,𝑦   𝑆,𝑎,𝑏,𝑐   𝑆,𝑔,𝑥   𝑇,𝑎,𝑏,𝑐   𝑇,𝑔,𝑥   𝑣,𝑢,𝑥,𝑦

Allowed substitution hints:   𝑆(𝑦,𝑣,𝑢)   𝑇(𝑦,𝑣,𝑢)   𝑂(𝑥,𝑣,𝑔,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑣,𝑢,𝑔,𝑎,𝑏,𝑐)   𝑊(𝑥,𝑦,𝑣,𝑢,𝑔,𝑎,𝑏,𝑐)

Proof of Theorem noetalem2

Step	Hyp	Ref	Expression
1		elex 3493	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2	1	anim2i 618	. . 3 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) → (𝐴 ⊆ No ∧ 𝐴 ∈ V))
3		elex 3493	. . . 4 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
4	3	anim2i 618	. . 3 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) → (𝐵 ⊆ No ∧ 𝐵 ∈ V))
5		id 22	. . 3 ⊢ (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏)
6	2, 4, 5	3anim123i 1152	. 2 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → ((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏))
7		noetalem2.1	. . . 4 ⊢ 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		noetalem2.2	. . . 4 ⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
9		eqid 2733	. . . 4 ⊢ (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o})) = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}))
10		eqid 2733	. . . 4 ⊢ (𝑇 ∪ ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o})) = (𝑇 ∪ ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}))
11	7, 8, 9, 10	noetalem1 27223	. . 3 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ((𝑆 ∈ No ∧ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏 ∧ ( bday ‘𝑆) ⊆ 𝑂)) ∨ (𝑇 ∈ No ∧ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏 ∧ ( bday ‘𝑇) ⊆ 𝑂))))
12		breq2 5150	. . . . . . 7 ⊢ (𝑐 = 𝑆 → (𝑎 <s 𝑐 ↔ 𝑎 <s 𝑆))
13	12	ralbidv 3178	. . . . . 6 ⊢ (𝑐 = 𝑆 → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑐 ↔ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑆))
14		breq1 5149	. . . . . . 7 ⊢ (𝑐 = 𝑆 → (𝑐 <s 𝑏 ↔ 𝑆 <s 𝑏))
15	14	ralbidv 3178	. . . . . 6 ⊢ (𝑐 = 𝑆 → (∀𝑏 ∈ 𝐵 𝑐 <s 𝑏 ↔ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏))
16		fveq2 6887	. . . . . . 7 ⊢ (𝑐 = 𝑆 → ( bday ‘𝑐) = ( bday ‘𝑆))
17	16	sseq1d 4011	. . . . . 6 ⊢ (𝑐 = 𝑆 → (( bday ‘𝑐) ⊆ 𝑂 ↔ ( bday ‘𝑆) ⊆ 𝑂))
18	13, 15, 17	3anbi123d 1437	. . . . 5 ⊢ (𝑐 = 𝑆 → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑐 ∧ ∀𝑏 ∈ 𝐵 𝑐 <s 𝑏 ∧ ( bday ‘𝑐) ⊆ 𝑂) ↔ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏 ∧ ( bday ‘𝑆) ⊆ 𝑂)))
19	18	rspcev 3611	. . . 4 ⊢ ((𝑆 ∈ No ∧ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏 ∧ ( bday ‘𝑆) ⊆ 𝑂)) → ∃𝑐 ∈ No (∀𝑎 ∈ 𝐴 𝑎 <s 𝑐 ∧ ∀𝑏 ∈ 𝐵 𝑐 <s 𝑏 ∧ ( bday ‘𝑐) ⊆ 𝑂))
20		breq2 5150	. . . . . . 7 ⊢ (𝑐 = 𝑇 → (𝑎 <s 𝑐 ↔ 𝑎 <s 𝑇))
21	20	ralbidv 3178	. . . . . 6 ⊢ (𝑐 = 𝑇 → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑐 ↔ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑇))
22		breq1 5149	. . . . . . 7 ⊢ (𝑐 = 𝑇 → (𝑐 <s 𝑏 ↔ 𝑇 <s 𝑏))
23	22	ralbidv 3178	. . . . . 6 ⊢ (𝑐 = 𝑇 → (∀𝑏 ∈ 𝐵 𝑐 <s 𝑏 ↔ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏))
24		fveq2 6887	. . . . . . 7 ⊢ (𝑐 = 𝑇 → ( bday ‘𝑐) = ( bday ‘𝑇))
25	24	sseq1d 4011	. . . . . 6 ⊢ (𝑐 = 𝑇 → (( bday ‘𝑐) ⊆ 𝑂 ↔ ( bday ‘𝑇) ⊆ 𝑂))
26	21, 23, 25	3anbi123d 1437	. . . . 5 ⊢ (𝑐 = 𝑇 → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑐 ∧ ∀𝑏 ∈ 𝐵 𝑐 <s 𝑏 ∧ ( bday ‘𝑐) ⊆ 𝑂) ↔ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏 ∧ ( bday ‘𝑇) ⊆ 𝑂)))
27	26	rspcev 3611	. . . 4 ⊢ ((𝑇 ∈ No ∧ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏 ∧ ( bday ‘𝑇) ⊆ 𝑂)) → ∃𝑐 ∈ No (∀𝑎 ∈ 𝐴 𝑎 <s 𝑐 ∧ ∀𝑏 ∈ 𝐵 𝑐 <s 𝑏 ∧ ( bday ‘𝑐) ⊆ 𝑂))
28	19, 27	jaoi 856	. . 3 ⊢ (((𝑆 ∈ No ∧ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏 ∧ ( bday ‘𝑆) ⊆ 𝑂)) ∨ (𝑇 ∈ No ∧ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏 ∧ ( bday ‘𝑇) ⊆ 𝑂))) → ∃𝑐 ∈ No (∀𝑎 ∈ 𝐴 𝑎 <s 𝑐 ∧ ∀𝑏 ∈ 𝐵 𝑐 <s 𝑏 ∧ ( bday ‘𝑐) ⊆ 𝑂))
29	11, 28	syl 17	. 2 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ∃𝑐 ∈ No (∀𝑎 ∈ 𝐴 𝑎 <s 𝑐 ∧ ∀𝑏 ∈ 𝐵 𝑐 <s 𝑏 ∧ ( bday ‘𝑐) ⊆ 𝑂))
30	6, 29	sylan 581	1 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ∃𝑐 ∈ No (∀𝑎 ∈ 𝐴 𝑎 <s 𝑐 ∧ ∀𝑏 ∈ 𝐵 𝑐 <s 𝑏 ∧ ( bday ‘𝑐) ⊆ 𝑂))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  ∀wral 3062  ∃wrex 3071  Vcvv 3475   ∖ cdif 3943   ∪ cun 3944   ⊆ wss 3946  ifcif 4526  {csn 4626  ⟨cop 4632  ∪ cuni 4906   class class class wbr 5146   ↦ cmpt 5229   × cxp 5672  dom cdm 5674   ↾ cres 5676   “ cima 5677  Oncon0 6360  suc csuc 6362  ℩cio 6489  ‘cfv 6539  ℩crio 7358  1_oc1o 8453  2_oc2o 8454   No csur 27122   <s cslt 27123   bday cbday 27124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5283  ax-sep 5297  ax-nul 5304  ax-pr 5425  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4907  df-int 4949  df-iun 4997  df-br 5147  df-opab 5209  df-mpt 5230  df-tr 5264  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7359  df-1o 8460  df-2o 8461  df-no 27125  df-slt 27126  df-bday 27127

This theorem is referenced by:  noeta  27225