Theorem noinfbnd1lem3 27791

Description: Lemma for noinfbnd1 27795. 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 27784	. . . . 5 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) → 𝑇 ∈ No )
3	2	3ad2ant2 1148	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → 𝑇 ∈ No )
4		nodmord 27719	. . . 4 ⊢ (𝑇 ∈ No → Ord dom 𝑇)
5		ordirr 6366	. . . 4 ⊢ (Ord dom 𝑇 → ¬ dom 𝑇 ∈ dom 𝑇)
6	3, 4, 5	3syl 18	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → ¬ dom 𝑇 ∈ dom 𝑇)
7		simpl3l 1243	. . . . 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 6901	. . . . . . . 8 ⊢ (¬ dom 𝑇 ∈ dom 𝑈 → (𝑈‘dom 𝑇) = ∅)
9		1n0 8458	. . . . . . . . . . 11 ⊢ 1o ≠ ∅
10	9	necomi 3013	. . . . . . . . . 10 ⊢ ∅ ≠ 1o
11		neeq1 3021	. . . . . . . . . 10 ⊢ ((𝑈‘dom 𝑇) = ∅ → ((𝑈‘dom 𝑇) ≠ 1o ↔ ∅ ≠ 1o))
12	10, 11	mpbiri 260	. . . . . . . . 9 ⊢ ((𝑈‘dom 𝑇) = ∅ → (𝑈‘dom 𝑇) ≠ 1o)
13	12	neneqd 2964	. . . . . . . 8 ⊢ ((𝑈‘dom 𝑇) = ∅ → ¬ (𝑈‘dom 𝑇) = 1o)
14	8, 13	syl 17	. . . . . . 7 ⊢ (¬ dom 𝑇 ∈ dom 𝑈 → ¬ (𝑈‘dom 𝑇) = 1o)
15	14	con4i 114	. . . . . 6 ⊢ ((𝑈‘dom 𝑇) = 1o → dom 𝑇 ∈ dom 𝑈)
16	15	adantl 485	. . . . 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 1241	. . . . . . . . . 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 484	. . . . . . . 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 484	. . . . . . . . 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 780	. . . . . . . . 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 484	. . . . . . . . . 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 27716	. . . . . . . . . 10 ⊢ (𝑇 ∈ No → dom 𝑇 ∈ On)
25	23, 24	syl 17	. . . . . . . . 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 484	. . . . . . . 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 1244	. . . . . . . . . 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 484	. . . . . . . . 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 1227	. . . . . . . . . 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 1228	. . . . . . . . . 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 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)) → (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇))
32		simpr 488	. . . . . . . . . 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 27790	. . . . . . . . . 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 1395	. . . . . . . . 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 2802	. . . . . . . 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 778	. . . . . . . 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 782	. . . . . . . 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 27740	. . . . . . . 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 1413	. . . . . . 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 460	. . . . . 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 3156	. . . . 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 5881	. . . . . . . 8 ⊢ (𝑝 = 𝑈 → dom 𝑝 = dom 𝑈)
43	42	eleq2d 2850	. . . . . . 7 ⊢ (𝑝 = 𝑈 → (dom 𝑇 ∈ dom 𝑝 ↔ dom 𝑇 ∈ dom 𝑈))
44		breq1 5105	. . . . . . . . . 10 ⊢ (𝑝 = 𝑈 → (𝑝 <s 𝑞 ↔ 𝑈 <s 𝑞))
45	44	notbid 320	. . . . . . . . 9 ⊢ (𝑝 = 𝑈 → (¬ 𝑝 <s 𝑞 ↔ ¬ 𝑈 <s 𝑞))
46		reseq1 5961	. . . . . . . . . 10 ⊢ (𝑝 = 𝑈 → (𝑝 ↾ suc dom 𝑇) = (𝑈 ↾ suc dom 𝑇))
47	46	eqeq1d 2766	. . . . . . . . 9 ⊢ (𝑝 = 𝑈 → ((𝑝 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇) ↔ (𝑈 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇)))
48	45, 47	imbi12d 346	. . . . . . . 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 3187	. . . . . . 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 641	. . . . . 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 847	. . . 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 27785	. . . . . . . 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 2850	. . . . . . 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 1147	. . . . . 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 2852	. . . . . . . . . 10 ⊢ (𝑧 = dom 𝑇 → (𝑧 ∈ dom 𝑝 ↔ dom 𝑇 ∈ dom 𝑝))
57		suceq 6416	. . . . . . . . . . . . . 14 ⊢ (𝑧 = dom 𝑇 → suc 𝑧 = suc dom 𝑇)
58	57	reseq2d 5967	. . . . . . . . . . . . 13 ⊢ (𝑧 = dom 𝑇 → (𝑝 ↾ suc 𝑧) = (𝑝 ↾ suc dom 𝑇))
59	57	reseq2d 5967	. . . . . . . . . . . . 13 ⊢ (𝑧 = dom 𝑇 → (𝑞 ↾ suc 𝑧) = (𝑞 ↾ suc dom 𝑇))
60	58, 59	eqeq12d 2780	. . . . . . . . . . . 12 ⊢ (𝑧 = dom 𝑇 → ((𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧) ↔ (𝑝 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇)))
61	60	imbi2d 342	. . . . . . . . . . 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 3187	. . . . . . . . . 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 641	. . . . . . . . 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 3188	. . . . . . . 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 18	. . . . . 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 281	. . . . 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 484	. . . 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 259	. . 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 825	. 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 2966	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 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  {cab 2742   ≠ wne 2959  ∀wral 3078  ∃wrex 3088   ∪ cun 3904   ⊆ wss 3906  ∅c0 4287  ifcif 4482  {csn 4584  ⟨cop 4590   class class class wbr 5102   ↦ cmpt 5183  dom cdm 5649   ↾ cres 5651  Ord word 6347  Oncon0 6348  suc csuc 6350  ℩cio 6477  ‘cfv 6523  ℩crio 7354  1_oc1o 8432   No csur 27706   <s clts 27707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ord 6351  df-on 6352  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-fo 6529  df-fv 6531  df-riota 7355  df-1o 8439  df-2o 8440  df-no 27709  df-lts 27710  df-bday 27711

This theorem is referenced by:  noinfbnd1lem4  27792  noinfbnd1lem5  27793  noinfbnd1lem6  27794