Theorem noinfbnd1lem3 27855

Description: Lemma for noinfbnd1 27859. 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 27848	. . . . 5 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) → 𝑇 ∈ No )
3	2	3ad2ant2 1150	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → 𝑇 ∈ No )
4		nodmord 27783	. . . 4 ⊢ (𝑇 ∈ No → Ord dom 𝑇)
5		ordirr 6379	. . . 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 1245	. . . . 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 6914	. . . . . . . 8 ⊢ (¬ dom 𝑇 ∈ dom 𝑈 → (𝑈‘dom 𝑇) = ∅)
9		1n0 8472	. . . . . . . . . . 11 ⊢ 1o ≠ ∅
10	9	necomi 3018	. . . . . . . . . 10 ⊢ ∅ ≠ 1o
11		neeq1 3026	. . . . . . . . . 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 2969	. . . . . . . 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 486	. . . . 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 1243	. . . . . . . . . 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 3946	. . . . . . . . 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 485	. . . . . . . 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 485	. . . . . . . . 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 782	. . . . . . . . 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 3946	. . . . . . . 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 485	. . . . . . . . . 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 27780	. . . . . . . . . 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 485	. . . . . . . 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 1246	. . . . . . . . . 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 485	. . . . . . . . 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 1229	. . . . . . . . . 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 1230	. . . . . . . . . 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 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)) → (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇))
32		simpr 489	. . . . . . . . . 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 27854	. . . . . . . . . 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 1399	. . . . . . . . 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 2807	. . . . . . . 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 780	. . . . . . . 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 784	. . . . . . . 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 27804	. . . . . . . 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 1417	. . . . . . 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 461	. . . . . 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 3163	. . . . 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 5894	. . . . . . . 8 ⊢ (𝑝 = 𝑈 → dom 𝑝 = dom 𝑈)
43	42	eleq2d 2855	. . . . . . 7 ⊢ (𝑝 = 𝑈 → (dom 𝑇 ∈ dom 𝑝 ↔ dom 𝑇 ∈ dom 𝑈))
44		breq1 5116	. . . . . . . . . 10 ⊢ (𝑝 = 𝑈 → (𝑝 <s 𝑞 ↔ 𝑈 <s 𝑞))
45	44	notbid 321	. . . . . . . . 9 ⊢ (𝑝 = 𝑈 → (¬ 𝑝 <s 𝑞 ↔ ¬ 𝑈 <s 𝑞))
46		reseq1 5973	. . . . . . . . . 10 ⊢ (𝑝 = 𝑈 → (𝑝 ↾ suc dom 𝑇) = (𝑈 ↾ suc dom 𝑇))
47	46	eqeq1d 2771	. . . . . . . . 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 3194	. . . . . . 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 643	. . . . . 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 3590	. . . . 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 849	. . . 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 27849	. . . . . . . 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 2855	. . . . . . 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 1149	. . . . . 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 2857	. . . . . . . . . 10 ⊢ (𝑧 = dom 𝑇 → (𝑧 ∈ dom 𝑝 ↔ dom 𝑇 ∈ dom 𝑝))
57		suceq 6430	. . . . . . . . . . . . . 14 ⊢ (𝑧 = dom 𝑇 → suc 𝑧 = suc dom 𝑇)
58	57	reseq2d 5979	. . . . . . . . . . . . 13 ⊢ (𝑧 = dom 𝑇 → (𝑝 ↾ suc 𝑧) = (𝑝 ↾ suc dom 𝑇))
59	57	reseq2d 5979	. . . . . . . . . . . . 13 ⊢ (𝑧 = dom 𝑇 → (𝑞 ↾ suc 𝑧) = (𝑞 ↾ suc dom 𝑇))
60	58, 59	eqeq12d 2785	. . . . . . . . . . . 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 3194	. . . . . . . . . 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 643	. . . . . . . . 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 3195	. . . . . . . 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 3644	. . . . . . 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 485	. . . 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 827	. 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 2971	1 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → (𝑈‘dom 𝑇) ≠ 1o)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  {cab 2747   ≠ wne 2964  ∀wral 3085  ∃wrex 3095   ∪ cun 3911   ⊆ wss 3913  ∅c0 4294  ifcif 4492  {csn 4594  ⟨cop 4600   class class class wbr 5113   ↦ cmpt 5196  dom cdm 5662   ↾ cres 5664  Ord word 6360  Oncon0 6361  suc csuc 6363  ℩cio 6491  ‘cfv 6537  ℩crio 7367  1_oc1o 8446   No csur 27770   <s clts 27771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-int 4917  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-riota 7368  df-1o 8453  df-2o 8454  df-no 27773  df-lts 27774  df-bday 27775

This theorem is referenced by:  noinfbnd1lem4  27856  noinfbnd1lem5  27857  noinfbnd1lem6  27858