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Theorem noetalem1 27727

Description: Lemma for noeta 27729. Either 𝑆 or 𝑇 satisfies the final condition. (Contributed by Scott Fenton, 9-Aug-2024.)

**Hypotheses**
Ref	Expression
noetalem1.1	⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}), (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))))
noetalem1.2	⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1_o⟩}), (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))))
noetalem1.3	⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1_o}))
noetalem1.4	⊢ 𝑊 = (𝑇 ∪ ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2_o}))

**Assertion**
Ref	Expression
noetalem1	⊢ ((((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))))

Distinct variable groups: 𝐴,𝑎,𝑏,𝑦 𝑍,𝑎,𝑏,𝑔,𝑥 𝑢,𝑂,𝑦 𝑊,𝑎,𝑏,𝑔,𝑥 𝐴,𝑔,𝑢,𝑣,𝑥,𝑎 𝐵,𝑎,𝑏,𝑔,𝑣,𝑥,𝑦 𝑇,𝑎,𝑏,𝑔,𝑥 𝑢,𝐵 𝑆,𝑎,𝑏,𝑔,𝑥 Allowed substitution hints: 𝑆(𝑦,𝑣,𝑢) 𝑇(𝑦,𝑣,𝑢) 𝑂(𝑥,𝑣,𝑔,𝑎,𝑏) 𝑊(𝑦,𝑣,𝑢) 𝑍(𝑦,𝑣,𝑢)

**Proof of Theorem noetalem1**
Step	Hyp	Ref	Expression
1		noetalem1.2	. . . . . . . . . 10 ⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1_o⟩}), (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))))
2	1	noinfno 27704	. . . . . . . . 9 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → 𝑇 ∈ No )
3	2	adantl 483	. . . . . . . 8 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → 𝑇 ∈ No )
4		nodmord 27639	. . . . . . . 8 ⊢ (𝑇 ∈ No → Ord dom 𝑇)
5	3, 4	syl 17	. . . . . . 7 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → Ord dom 𝑇)
6		noetalem1.1	. . . . . . . . . 10 ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}), (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))))
7	6	nosupno 27689	. . . . . . . . 9 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
8	7	adantr 482	. . . . . . . 8 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → 𝑆 ∈ No )
9		nodmord 27639	. . . . . . . 8 ⊢ (𝑆 ∈ No → Ord dom 𝑆)
10	8, 9	syl 17	. . . . . . 7 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → Ord dom 𝑆)
11		ordtri2or2 6415	. . . . . . 7 ⊢ ((Ord dom 𝑇 ∧ Ord dom 𝑆) → (dom 𝑇 ⊆ dom 𝑆 ∨ dom 𝑆 ⊆ dom 𝑇))
12	5, 10, 11	syl2anc 591	. . . . . 6 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → (dom 𝑇 ⊆ dom 𝑆 ∨ dom 𝑆 ⊆ dom 𝑇))
13		ssequn2 4121	. . . . . . 7 ⊢ (dom 𝑇 ⊆ dom 𝑆 ↔ (dom 𝑆 ∪ dom 𝑇) = dom 𝑆)
14		ssequn1 4118	. . . . . . 7 ⊢ (dom 𝑆 ⊆ dom 𝑇 ↔ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇)
15	13, 14	orbi12i 921	. . . . . 6 ⊢ ((dom 𝑇 ⊆ dom 𝑆 ∨ dom 𝑆 ⊆ dom 𝑇) ↔ ((dom 𝑆 ∪ dom 𝑇) = dom 𝑆 ∨ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇))
16	12, 15	sylib 220	. . . . 5 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → ((dom 𝑆 ∪ dom 𝑇) = dom 𝑆 ∨ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇))
17	16	3adant3 1139	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → ((dom 𝑆 ∪ dom 𝑇) = dom 𝑆 ∨ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇))
18		simplll 781	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ 𝑎 ∈ 𝐴) → 𝐴 ⊆ No )
19		simpllr 782	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ 𝑎 ∈ 𝐴) → 𝐴 ∈ V)
20		simplrr 784	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ 𝑎 ∈ 𝐴) → 𝐵 ∈ V)
21		simpr 486	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
22		noetalem1.3	. . . . . . . . . . . . 13 ⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1_o}))
23	6, 22	noetasuplem3 27721	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑎 ∈ 𝐴) → 𝑎 <s 𝑍)
24	18, 19, 20, 21, 23	syl31anc 1382	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ 𝑎 ∈ 𝐴) → 𝑎 <s 𝑍)
25	24	ralrimiva 3133	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)
26	25	3adant3 1139	. . . . . . . . 9 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)
27	6, 22	noetasuplem4 27722	. . . . . . . . 9 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏)
28	26, 27	jca 517	. . . . . . . 8 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑍 ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏))
29	28	adantr 482	. . . . . . 7 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑆) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑍 ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏))
30		simp1l 1205	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → 𝐴 ⊆ No )
31		simp1r 1206	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → 𝐴 ∈ V)
32		simp2r 1208	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → 𝐵 ∈ V)
33	6, 22	noetasuplem1 27719	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑍 ∈ No )
34	30, 31, 32, 33	syl3anc 1380	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → 𝑍 ∈ No )
35	6, 1	nosupinfsep 27718	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑍 ∈ No ) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑍 ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ↔ (∀𝑎 ∈ 𝐴 𝑎 <s (𝑍 ↾ (dom 𝑆 ∪ dom 𝑇)) ∧ ∀𝑏 ∈ 𝐵 (𝑍 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏)))
36	34, 35	syld3an3 1418	. . . . . . . . 9 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑍 ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ↔ (∀𝑎 ∈ 𝐴 𝑎 <s (𝑍 ↾ (dom 𝑆 ∪ dom 𝑇)) ∧ ∀𝑏 ∈ 𝐵 (𝑍 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏)))
37	36	adantr 482	. . . . . . . 8 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑆) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑍 ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ↔ (∀𝑎 ∈ 𝐴 𝑎 <s (𝑍 ↾ (dom 𝑆 ∪ dom 𝑇)) ∧ ∀𝑏 ∈ 𝐵 (𝑍 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏)))
38		simpr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑆) → (dom 𝑆 ∪ dom 𝑇) = dom 𝑆)
39	38	reseq2d 5938	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑆) → (𝑍 ↾ (dom 𝑆 ∪ dom 𝑇)) = (𝑍 ↾ dom 𝑆))
40		simplll 781	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑆) → 𝐴 ⊆ No )
41		simpllr 782	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑆) → 𝐴 ∈ V)
42		simplrr 784	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑆) → 𝐵 ∈ V)
43	6, 22	noetasuplem2 27720	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑍 ↾ dom 𝑆) = 𝑆)
44	40, 41, 42, 43	syl3anc 1380	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑆) → (𝑍 ↾ dom 𝑆) = 𝑆)
45	39, 44	eqtrd 2776	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑆) → (𝑍 ↾ (dom 𝑆 ∪ dom 𝑇)) = 𝑆)
46	45	breq2d 5087	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑆) → (𝑎 <s (𝑍 ↾ (dom 𝑆 ∪ dom 𝑇)) ↔ 𝑎 <s 𝑆))
47	46	ralbidv 3164	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑆) → (∀𝑎 ∈ 𝐴 𝑎 <s (𝑍 ↾ (dom 𝑆 ∪ dom 𝑇)) ↔ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑆))
48	45	breq1d 5085	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑆) → ((𝑍 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏 ↔ 𝑆 <s 𝑏))
49	48	ralbidv 3164	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑆) → (∀𝑏 ∈ 𝐵 (𝑍 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏 ↔ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏))
50	47, 49	anbi12d 639	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑆) → ((∀𝑎 ∈ 𝐴 𝑎 <s (𝑍 ↾ (dom 𝑆 ∪ dom 𝑇)) ∧ ∀𝑏 ∈ 𝐵 (𝑍 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏) ↔ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏)))
51	50	3adantl3 1176	. . . . . . . 8 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑆) → ((∀𝑎 ∈ 𝐴 𝑎 <s (𝑍 ↾ (dom 𝑆 ∪ dom 𝑇)) ∧ ∀𝑏 ∈ 𝐵 (𝑍 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏) ↔ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏)))
52	37, 51	bitrd 281	. . . . . . 7 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑆) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑍 ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ↔ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏)))
53	29, 52	mpbid 234	. . . . . 6 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑆) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏))
54	53	ex 414	. . . . 5 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → ((dom 𝑆 ∪ dom 𝑇) = dom 𝑆 → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏)))
55		noetalem1.4	. . . . . . . . . 10 ⊢ 𝑊 = (𝑇 ∪ ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2_o}))
56	1, 55	noetainflem4 27726	. . . . . . . . 9 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → ∀𝑎 ∈ 𝐴 𝑎 <s 𝑊)
57		simpllr 782	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ 𝑏 ∈ 𝐵) → 𝐴 ∈ V)
58		simplrl 783	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ 𝑏 ∈ 𝐵) → 𝐵 ⊆ No )
59		simplrr 784	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ 𝑏 ∈ 𝐵) → 𝐵 ∈ V)
60		simpr 486	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
61	1, 55	noetainflem3 27725	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ V ∧ 𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑏 ∈ 𝐵) → 𝑊 <s 𝑏)
62	57, 58, 59, 60, 61	syl31anc 1382	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ 𝑏 ∈ 𝐵) → 𝑊 <s 𝑏)
63	62	ralrimiva 3133	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → ∀𝑏 ∈ 𝐵 𝑊 <s 𝑏)
64	63	3adant3 1139	. . . . . . . . 9 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → ∀𝑏 ∈ 𝐵 𝑊 <s 𝑏)
65	56, 64	jca 517	. . . . . . . 8 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑊 ∧ ∀𝑏 ∈ 𝐵 𝑊 <s 𝑏))
66	65	adantr 482	. . . . . . 7 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑊 ∧ ∀𝑏 ∈ 𝐵 𝑊 <s 𝑏))
67		simpl1 1199	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇) → (𝐴 ⊆ No ∧ 𝐴 ∈ V))
68		simpl2l 1234	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇) → 𝐵 ⊆ No )
69		simpl2r 1235	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇) → 𝐵 ∈ V)
70		simpl1r 1233	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇) → 𝐴 ∈ V)
71	1, 55	noetainflem1 27723	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ 𝐵 ⊆ No ∧ 𝐵 ∈ V) → 𝑊 ∈ No )
72	70, 68, 69, 71	syl3anc 1380	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇) → 𝑊 ∈ No )
73	6, 1	nosupinfsep 27718	. . . . . . . . 9 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑊 ∧ ∀𝑏 ∈ 𝐵 𝑊 <s 𝑏) ↔ (∀𝑎 ∈ 𝐴 𝑎 <s (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∧ ∀𝑏 ∈ 𝐵 (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏)))
74	67, 68, 69, 72, 73	syl121anc 1384	. . . . . . . 8 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑊 ∧ ∀𝑏 ∈ 𝐵 𝑊 <s 𝑏) ↔ (∀𝑎 ∈ 𝐴 𝑎 <s (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∧ ∀𝑏 ∈ 𝐵 (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏)))
75		simpr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇) → (dom 𝑆 ∪ dom 𝑇) = dom 𝑇)
76	75	reseq2d 5938	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇) → (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) = (𝑊 ↾ dom 𝑇))
77		simplr 775	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇) → (𝐵 ⊆ No ∧ 𝐵 ∈ V))
78	1, 55	noetainflem2 27724	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → (𝑊 ↾ dom 𝑇) = 𝑇)
79	77, 78	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇) → (𝑊 ↾ dom 𝑇) = 𝑇)
80	76, 79	eqtrd 2776	. . . . . . . . . . . 12 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇) → (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) = 𝑇)
81	80	breq2d 5087	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇) → (𝑎 <s (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↔ 𝑎 <s 𝑇))
82	81	ralbidv 3164	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇) → (∀𝑎 ∈ 𝐴 𝑎 <s (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ↔ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑇))
83	80	breq1d 5085	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇) → ((𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏 ↔ 𝑇 <s 𝑏))
84	83	ralbidv 3164	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇) → (∀𝑏 ∈ 𝐵 (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏 ↔ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏))
85	82, 84	anbi12d 639	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇) → ((∀𝑎 ∈ 𝐴 𝑎 <s (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∧ ∀𝑏 ∈ 𝐵 (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏) ↔ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏)))
86	85	3adantl3 1176	. . . . . . . 8 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇) → ((∀𝑎 ∈ 𝐴 𝑎 <s (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∧ ∀𝑏 ∈ 𝐵 (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏) ↔ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏)))
87	74, 86	bitrd 281	. . . . . . 7 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑊 ∧ ∀𝑏 ∈ 𝐵 𝑊 <s 𝑏) ↔ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏)))
88	66, 87	mpbid 234	. . . . . 6 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏))
89	88	ex 414	. . . . 5 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → ((dom 𝑆 ∪ dom 𝑇) = dom 𝑇 → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏)))
90	54, 89	orim12d 973	. . . 4 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → (((dom 𝑆 ∪ dom 𝑇) = dom 𝑆 ∨ (dom 𝑆 ∪ dom 𝑇) = dom 𝑇) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏) ∨ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏))))
91	17, 90	mpd 15	. . 3 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏) ∨ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏)))
92	91	adantr 482	. 2 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏) ∨ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏)))
93		simpll 773	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → (𝐴 ⊆ No ∧ 𝐴 ∈ V))
94		simprl 777	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → 𝑂 ∈ On)
95		ssun1 4110	. . . . . . . . . . 11 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
96		imass2 6061	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ (𝐴 ∪ 𝐵) → ( bday “ 𝐴) ⊆ ( bday “ (𝐴 ∪ 𝐵)))
97	95, 96	ax-mp 5	. . . . . . . . . 10 ⊢ ( bday “ 𝐴) ⊆ ( bday “ (𝐴 ∪ 𝐵))
98		simprr 779	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)
99	97, 98	sstrid 3928	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ( bday “ 𝐴) ⊆ 𝑂)
100	6	nosupbday 27691	. . . . . . . . 9 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday “ 𝐴) ⊆ 𝑂)) → ( bday ‘𝑆) ⊆ 𝑂)
101	93, 94, 99, 100	syl12anc 843	. . . . . . . 8 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ( bday ‘𝑆) ⊆ 𝑂)
102	101	a1d 25	. . . . . . 7 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏) → ( bday ‘𝑆) ⊆ 𝑂))
103	102	ancld 556	. . . . . 6 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏) ∧ ( bday ‘𝑆) ⊆ 𝑂)))
104		df-3an 1095	. . . . . 6 ⊢ ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏 ∧ ( bday ‘𝑆) ⊆ 𝑂) ↔ ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏) ∧ ( bday ‘𝑆) ⊆ 𝑂))
105	103, 104	imbitrrdi 254	. . . . 5 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏 ∧ ( bday ‘𝑆) ⊆ 𝑂)))
106	93, 7	syl 17	. . . . 5 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → 𝑆 ∈ No )
107	105, 106	jctild 531	. . . 4 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏) → (𝑆 ∈ No ∧ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏 ∧ ( bday ‘𝑆) ⊆ 𝑂))))
108		simplr 775	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → (𝐵 ⊆ No ∧ 𝐵 ∈ V))
109		ssun2 4111	. . . . . . . . . . 11 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
110		imass2 6061	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ (𝐴 ∪ 𝐵) → ( bday “ 𝐵) ⊆ ( bday “ (𝐴 ∪ 𝐵)))
111	109, 110	ax-mp 5	. . . . . . . . . 10 ⊢ ( bday “ 𝐵) ⊆ ( bday “ (𝐴 ∪ 𝐵))
112	111, 98	sstrid 3928	. . . . . . . . 9 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ( bday “ 𝐵) ⊆ 𝑂)
113	1	noinfbday 27706	. . . . . . . . 9 ⊢ (((𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ (𝑂 ∈ On ∧ ( bday “ 𝐵) ⊆ 𝑂)) → ( bday ‘𝑇) ⊆ 𝑂)
114	108, 94, 112, 113	syl12anc 843	. . . . . . . 8 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ( bday ‘𝑇) ⊆ 𝑂)
115	114	a1d 25	. . . . . . 7 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏) → ( bday ‘𝑇) ⊆ 𝑂))
116	115	ancld 556	. . . . . 6 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏) ∧ ( bday ‘𝑇) ⊆ 𝑂)))
117		df-3an 1095	. . . . . 6 ⊢ ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏 ∧ ( bday ‘𝑇) ⊆ 𝑂) ↔ ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏) ∧ ( bday ‘𝑇) ⊆ 𝑂))
118	116, 117	imbitrrdi 254	. . . . 5 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏 ∧ ( bday ‘𝑇) ⊆ 𝑂)))
119	108, 2	syl 17	. . . . 5 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → 𝑇 ∈ No )
120	118, 119	jctild 531	. . . 4 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏) → (𝑇 ∈ No ∧ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏 ∧ ( bday ‘𝑇) ⊆ 𝑂))))
121	107, 120	orim12d 973	. . 3 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → (((∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏) ∨ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏)) → ((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)))))
122	121	3adantl3 1176	. 2 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → (((∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏) ∨ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏)) → ((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)))))
123	92, 122	mpd 15	1 ⊢ ((((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))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 397 ∨ wo 854 ∧ w3a 1093 = wceq 1548 ∈ wcel 2121 {cab 2719 ∀wral 3055 ∃wrex 3065 Vcvv 3433 ∖ cdif 3882 ∪ cun 3883 ⊆ wss 3885 ifcif 4457 {csn 4558 ⟨cop 4564 ∪ cuni 4841 class class class wbr 5075 ↦ cmpt 5156 × cxp 5619 dom cdm 5621 ↾ cres 5623 “ cima 5624 Ord word 6313 Oncon0 6314 suc csuc 6316 ℩cio 6443 ‘cfv 6489 ℩crio 7316 1_oc1o 8392 2_oc2o 8393 No csur 27625 <s clts 27626 bday cbday 27627

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-int 4881 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-ord 6317 df-on 6318 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-1o 8399 df-2o 8400 df-no 27628 df-lts 27629 df-bday 27630

This theorem is referenced by: noetalem2 27728

W3C validator