Theorem noinfbnd1lem3 27944

Description: Lemma for noinfbnd1 27948. If 𝑈 is a prolongment of 𝑇 and in 𝐵, then (𝑈‘dom 𝑇) is not 1_o. (Contributed by Scott Fenton, 9-Aug-2024.)

Hypothesis

Ref	Expression
noinfbnd1.1	⊢ 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
noinfbnd1lem3	⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → (𝑈‘dom 𝑇) ≠ 1o)

Distinct variable groups:   𝐵,𝑔,𝑢,𝑣,𝑥,𝑦   𝑣,𝑈   𝑔,𝑉

Allowed substitution hints:   𝑇(𝑥, 𝑦, 𝑣, 𝑢, 𝑔)   𝑈(𝑥, 𝑦, 𝑢, 𝑔)   𝑉(𝑥, 𝑦, 𝑣, 𝑢)

Proof of Theorem noinfbnd1lem3

Dummy variables 𝑝 𝑞 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		noinfbnd1.1	. . . . . 6 ⊢ 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))))
2	1	noinfno 27937	. . . . 5 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) → 𝑇 ∈ No )
3	2	3ad2ant2 1152	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → 𝑇 ∈ No )
4		nodmord 27872	. . . 4 ⊢ (𝑇 ∈ No → Ord dom 𝑇)
5		ordirr 6382	. . . 4 ⊢ (Ord dom 𝑇 → ¬ dom 𝑇 ∈ dom 𝑇)
6	3, 4, 5	3syl 19	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → ¬ dom 𝑇 ∈ dom 𝑇)
7		simpl3l 1247	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) → 𝑈 ∈ 𝐵)
8		ndmfv 6917	. . . . . . . 8 ⊢ (¬ dom 𝑇 ∈ dom 𝑈 → (𝑈‘dom 𝑇) = ∅)
9		1n0 8478	. . . . . . . . . . 11 ⊢ 1o ≠ ∅
10	9	necomi 3014	. . . . . . . . . 10 ⊢ ∅ ≠ 1o
11		neeq1 3022	. . . . . . . . . 10 ⊢ ((𝑈‘dom 𝑇) = ∅ → ((𝑈‘dom 𝑇) ≠ 1o ↔ ∅ ≠ 1o))
12	10, 11	mpbiri 261	. . . . . . . . 9 ⊢ ((𝑈‘dom 𝑇) = ∅ → (𝑈‘dom 𝑇) ≠ 1o)
13	12	neneqd 2965	. . . . . . . 8 ⊢ ((𝑈‘dom 𝑇) = ∅ → ¬ (𝑈‘dom 𝑇) = 1o)
14	8, 13	syl 18	. . . . . . 7 ⊢ (¬ dom 𝑇 ∈ dom 𝑈 → ¬ (𝑈‘dom 𝑇) = 1o)
15	14	con4i 115	. . . . . 6 ⊢ ((𝑈‘dom 𝑇) = 1o → dom 𝑇 ∈ dom 𝑈)
16	15	adantl 487	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) → dom 𝑇 ∈ dom 𝑈)
17		simpl2l 1245	. . . . . . . . . 10 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) → 𝐵 ⊆ No )
18	17, 7	sseldd 3939	. . . . . . . . 9 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) → 𝑈 ∈ No )
19	18	adantr 486	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → 𝑈 ∈ No )
20	17	adantr 486	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → 𝐵 ⊆ No )
21		simprl 783	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → 𝑞 ∈ 𝐵)
22	20, 21	sseldd 3939	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → 𝑞 ∈ No )
23	3	adantr 486	. . . . . . . . . 10 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) → 𝑇 ∈ No )
24		nodmon 27869	. . . . . . . . . 10 ⊢ (𝑇 ∈ No → dom 𝑇 ∈ On)
25	23, 24	syl 18	. . . . . . . . 9 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) → dom 𝑇 ∈ On)
26	25	adantr 486	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → dom 𝑇 ∈ On)
27		simpl3r 1248	. . . . . . . . . 10 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) → (𝑈 ↾ dom 𝑇) = 𝑇)
28	27	adantr 486	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → (𝑈 ↾ dom 𝑇) = 𝑇)
29		simpll1 1231	. . . . . . . . . 10 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → ¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥)
30		simpll2 1232	. . . . . . . . . 10 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉))
31		simpll3 1233	. . . . . . . . . 10 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇))
32		simpr 490	. . . . . . . . . 10 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞))
33	1	noinfbnd1lem2 27943	. . . . . . . . . 10 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ ((𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞))) → (𝑞 ↾ dom 𝑇) = 𝑇)
34	29, 30, 31, 32, 33	syl112anc 1401	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → (𝑞 ↾ dom 𝑇) = 𝑇)
35	28, 34	eqtr4d 2803	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → (𝑈 ↾ dom 𝑇) = (𝑞 ↾ dom 𝑇))
36		simplr 781	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → (𝑈‘dom 𝑇) = 1o)
37		simprr 785	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → ¬ 𝑈 <s 𝑞)
38		nogesgn1ores 27893	. . . . . . . 8 ⊢ (((𝑈 ∈ No ∧ 𝑞 ∈ No ∧ dom 𝑇 ∈ On) ∧ ((𝑈 ↾ dom 𝑇) = (𝑞 ↾ dom 𝑇) ∧ (𝑈‘dom 𝑇) = 1o) ∧ ¬ 𝑈 <s 𝑞) → (𝑈 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇))
39	19, 22, 26, 35, 36, 37, 38	syl321anc 1419	. . . . . . 7 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → (𝑈 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇))
40	39	expr 462	. . . . . 6 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) ∧ 𝑞 ∈ 𝐵) → (¬ 𝑈 <s 𝑞 → (𝑈 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇)))
41	40	ralrimiva 3159	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) → ∀𝑞 ∈ 𝐵 (¬ 𝑈 <s 𝑞 → (𝑈 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇)))
42		dmeq 5895	. . . . . . . 8 ⊢ (𝑝 = 𝑈 → dom 𝑝 = dom 𝑈)
43	42	eleq2d 2851	. . . . . . 7 ⊢ (𝑝 = 𝑈 → (dom 𝑇 ∈ dom 𝑝 ↔ dom 𝑇 ∈ dom 𝑈))
44		breq1 5114	. . . . . . . . . 10 ⊢ (𝑝 = 𝑈 → (𝑝 <s 𝑞 ↔ 𝑈 <s 𝑞))
45	44	notbid 321	. . . . . . . . 9 ⊢ (𝑝 = 𝑈 → (¬ 𝑝 <s 𝑞 ↔ ¬ 𝑈 <s 𝑞))
46		reseq1 5974	. . . . . . . . . 10 ⊢ (𝑝 = 𝑈 → (𝑝 ↾ suc dom 𝑇) = (𝑈 ↾ suc dom 𝑇))
47	46	eqeq1d 2767	. . . . . . . . 9 ⊢ (𝑝 = 𝑈 → ((𝑝 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇) ↔ (𝑈 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇)))
48	45, 47	imbi12d 347	. . . . . . . 8 ⊢ (𝑝 = 𝑈 → ((¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇)) ↔ (¬ 𝑈 <s 𝑞 → (𝑈 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇))))
49	48	ralbidv 3190	. . . . . . 7 ⊢ (𝑝 = 𝑈 → (∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇)) ↔ ∀𝑞 ∈ 𝐵 (¬ 𝑈 <s 𝑞 → (𝑈 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇))))
50	43, 49	anbi12d 644	. . . . . 6 ⊢ (𝑝 = 𝑈 → ((dom 𝑇 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇))) ↔ (dom 𝑇 ∈ dom 𝑈 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑈 <s 𝑞 → (𝑈 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇)))))
51	50	rspcev 3583	. . . . 5 ⊢ ((𝑈 ∈ 𝐵 ∧ (dom 𝑇 ∈ dom 𝑈 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑈 <s 𝑞 → (𝑈 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇)))) → ∃𝑝 ∈ 𝐵 (dom 𝑇 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇))))
52	7, 16, 41, 51	syl12anc 850	. . . 4 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) → ∃𝑝 ∈ 𝐵 (dom 𝑇 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇))))
53	1	noinfdm 27938	. . . . . . . 8 ⊢ (¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = {𝑧 ∣ ∃𝑝 ∈ 𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))})
54	53	eleq2d 2851	. . . . . . 7 ⊢ (¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 → (dom 𝑇 ∈ dom 𝑇 ↔ dom 𝑇 ∈ {𝑧 ∣ ∃𝑝 ∈ 𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))}))
55	54	3ad2ant1 1151	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → (dom 𝑇 ∈ dom 𝑇 ↔ dom 𝑇 ∈ {𝑧 ∣ ∃𝑝 ∈ 𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))}))
56		eleq1 2853	. . . . . . . . . 10 ⊢ (𝑧 = dom 𝑇 → (𝑧 ∈ dom 𝑝 ↔ dom 𝑇 ∈ dom 𝑝))
57		suceq 6433	. . . . . . . . . . . . . 14 ⊢ (𝑧 = dom 𝑇 → suc 𝑧 = suc dom 𝑇)
58	57	reseq2d 5980	. . . . . . . . . . . . 13 ⊢ (𝑧 = dom 𝑇 → (𝑝 ↾ suc 𝑧) = (𝑝 ↾ suc dom 𝑇))
59	57	reseq2d 5980	. . . . . . . . . . . . 13 ⊢ (𝑧 = dom 𝑇 → (𝑞 ↾ suc 𝑧) = (𝑞 ↾ suc dom 𝑇))
60	58, 59	eqeq12d 2781	. . . . . . . . . . . 12 ⊢ (𝑧 = dom 𝑇 → ((𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧) ↔ (𝑝 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇)))
61	60	imbi2d 343	. . . . . . . . . . 11 ⊢ (𝑧 = dom 𝑇 → ((¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)) ↔ (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇))))
62	61	ralbidv 3190	. . . . . . . . . 10 ⊢ (𝑧 = dom 𝑇 → (∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)) ↔ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇))))
63	56, 62	anbi12d 644	. . . . . . . . 9 ⊢ (𝑧 = dom 𝑇 → ((𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧))) ↔ (dom 𝑇 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇)))))
64	63	rexbidv 3191	. . . . . . . 8 ⊢ (𝑧 = dom 𝑇 → (∃𝑝 ∈ 𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧))) ↔ ∃𝑝 ∈ 𝐵 (dom 𝑇 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇)))))
65	64	elabg 3637	. . . . . . 7 ⊢ (dom 𝑇 ∈ On → (dom 𝑇 ∈ {𝑧 ∣ ∃𝑝 ∈ 𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))} ↔ ∃𝑝 ∈ 𝐵 (dom 𝑇 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇)))))
66	3, 24, 65	3syl 19	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → (dom 𝑇 ∈ {𝑧 ∣ ∃𝑝 ∈ 𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))} ↔ ∃𝑝 ∈ 𝐵 (dom 𝑇 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇)))))
67	55, 66	bitrd 282	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → (dom 𝑇 ∈ dom 𝑇 ↔ ∃𝑝 ∈ 𝐵 (dom 𝑇 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇)))))
68	67	adantr 486	. . . 4 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) → (dom 𝑇 ∈ dom 𝑇 ↔ ∃𝑝 ∈ 𝐵 (dom 𝑇 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇)))))
69	52, 68	mpbird 260	. . 3 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1o) → dom 𝑇 ∈ dom 𝑇)
70	6, 69	mtand 828	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → ¬ (𝑈‘dom 𝑇) = 1o)
71	70	neqned 2967	1 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → (𝑈‘dom 𝑇) ≠ 1o)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  {cab 2743   ≠ wne 2960  ∀wral 3081  ∃wrex 3091   ∪ cun 3904   ⊆ wss 3906  ∅c0 4286  ifcif 4489  {csn 4591  ⟨cop 4597   class class class wbr 5111   ↦ cmpt 5194  dom cdm 5663   ↾ cres 5665  Ord word 6363  Oncon0 6364  suc csuc 6366  ℩cio 6494  ‘cfv 6540  ℩crio 7375  1_oc1o 8452   No csur 27859   <s clts 27860

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-int 4915  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-riota 7376  df-1o 8459  df-2o 8460  df-no 27862  df-lts 27863  df-bday 27864

This theorem is used by:  noinfbnd1lem4  27945  noinfbnd1lem5  27946  noinfbnd1lem6  27947