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Theorem noinfbnd1lem3 27217

Description: Lemma for noinfbnd1 27221. 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), 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))))

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

Distinct variable groups: 𝐵,𝑔,𝑢,𝑣,𝑥,𝑦 𝑣,𝑈 𝑥,𝑢,𝑦 𝑔,𝑉 𝑥,𝑣,𝑦 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), 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 27210	. . . . 5 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) → 𝑇 ∈ No )
3	2	3ad2ant2 1134	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → 𝑇 ∈ No )
4		nodmord 27145	. . . 4 ⊢ (𝑇 ∈ No → Ord dom 𝑇)
5		ordirr 6379	. . . 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 1228	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) → 𝑈 ∈ 𝐵)
8		ndmfv 6923	. . . . . . . 8 ⊢ (¬ dom 𝑇 ∈ dom 𝑈 → (𝑈‘dom 𝑇) = ∅)
9		1n0 8484	. . . . . . . . . . 11 ⊢ 1_o ≠ ∅
10	9	necomi 2995	. . . . . . . . . 10 ⊢ ∅ ≠ 1_o
11		neeq1 3003	. . . . . . . . . 10 ⊢ ((𝑈‘dom 𝑇) = ∅ → ((𝑈‘dom 𝑇) ≠ 1_o ↔ ∅ ≠ 1_o))
12	10, 11	mpbiri 257	. . . . . . . . 9 ⊢ ((𝑈‘dom 𝑇) = ∅ → (𝑈‘dom 𝑇) ≠ 1_o)
13	12	neneqd 2945	. . . . . . . 8 ⊢ ((𝑈‘dom 𝑇) = ∅ → ¬ (𝑈‘dom 𝑇) = 1_o)
14	8, 13	syl 17	. . . . . . 7 ⊢ (¬ dom 𝑇 ∈ dom 𝑈 → ¬ (𝑈‘dom 𝑇) = 1_o)
15	14	con4i 114	. . . . . 6 ⊢ ((𝑈‘dom 𝑇) = 1_o → dom 𝑇 ∈ dom 𝑈)
16	15	adantl 482	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) → dom 𝑇 ∈ dom 𝑈)
17		simpl2l 1226	. . . . . . . . . 10 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) → 𝐵 ⊆ No )
18	17, 7	sseldd 3982	. . . . . . . . 9 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) → 𝑈 ∈ No )
19	18	adantr 481	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → 𝑈 ∈ No )
20	17	adantr 481	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → 𝐵 ⊆ No )
21		simprl 769	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → 𝑞 ∈ 𝐵)
22	20, 21	sseldd 3982	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → 𝑞 ∈ No )
23	3	adantr 481	. . . . . . . . . 10 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) → 𝑇 ∈ No )
24		nodmon 27142	. . . . . . . . . 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) = 1_o) → dom 𝑇 ∈ On)
26	25	adantr 481	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → dom 𝑇 ∈ On)
27		simpl3r 1229	. . . . . . . . . 10 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) → (𝑈 ↾ dom 𝑇) = 𝑇)
28	27	adantr 481	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → (𝑈 ↾ dom 𝑇) = 𝑇)
29		simpll1 1212	. . . . . . . . . 10 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → ¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥)
30		simpll2 1213	. . . . . . . . . 10 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉))
31		simpll3 1214	. . . . . . . . . 10 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇))
32		simpr 485	. . . . . . . . . 10 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞))
33	1	noinfbnd1lem2 27216	. . . . . . . . . 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 1374	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → (𝑞 ↾ dom 𝑇) = 𝑇)
35	28, 34	eqtr4d 2775	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → (𝑈 ↾ dom 𝑇) = (𝑞 ↾ dom 𝑇))
36		simplr 767	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → (𝑈‘dom 𝑇) = 1_o)
37		simprr 771	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → ¬ 𝑈 <s 𝑞)
38		nogesgn1ores 27166	. . . . . . . 8 ⊢ (((𝑈 ∈ No ∧ 𝑞 ∈ No ∧ dom 𝑇 ∈ On) ∧ ((𝑈 ↾ dom 𝑇) = (𝑞 ↾ dom 𝑇) ∧ (𝑈‘dom 𝑇) = 1_o) ∧ ¬ 𝑈 <s 𝑞) → (𝑈 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇))
39	19, 22, 26, 35, 36, 37, 38	syl321anc 1392	. . . . . . 7 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) ∧ (𝑞 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑞)) → (𝑈 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇))
40	39	expr 457	. . . . . 6 ⊢ ((((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) ∧ 𝑞 ∈ 𝐵) → (¬ 𝑈 <s 𝑞 → (𝑈 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇)))
41	40	ralrimiva 3146	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) → ∀𝑞 ∈ 𝐵 (¬ 𝑈 <s 𝑞 → (𝑈 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇)))
42		dmeq 5901	. . . . . . . 8 ⊢ (𝑝 = 𝑈 → dom 𝑝 = dom 𝑈)
43	42	eleq2d 2819	. . . . . . 7 ⊢ (𝑝 = 𝑈 → (dom 𝑇 ∈ dom 𝑝 ↔ dom 𝑇 ∈ dom 𝑈))
44		breq1 5150	. . . . . . . . . 10 ⊢ (𝑝 = 𝑈 → (𝑝 <s 𝑞 ↔ 𝑈 <s 𝑞))
45	44	notbid 317	. . . . . . . . 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 2734	. . . . . . . . 9 ⊢ (𝑝 = 𝑈 → ((𝑝 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇) ↔ (𝑈 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇)))
48	45, 47	imbi12d 344	. . . . . . . 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 3177	. . . . . . 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 631	. . . . . 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 3612	. . . . 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 835	. . . 4 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) → ∃𝑝 ∈ 𝐵 (dom 𝑇 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇))))
53	1	noinfdm 27211	. . . . . . . 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 2819	. . . . . . 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 1133	. . . . . 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 2821	. . . . . . . . . 10 ⊢ (𝑧 = dom 𝑇 → (𝑧 ∈ dom 𝑝 ↔ dom 𝑇 ∈ dom 𝑝))
57		suceq 6427	. . . . . . . . . . . . . 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 2748	. . . . . . . . . . . 12 ⊢ (𝑧 = dom 𝑇 → ((𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧) ↔ (𝑝 ↾ suc dom 𝑇) = (𝑞 ↾ suc dom 𝑇)))
61	60	imbi2d 340	. . . . . . . . . . 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 3177	. . . . . . . . . 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 631	. . . . . . . . 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 3178	. . . . . . . 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 3665	. . . . . . 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 278	. . . . 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 481	. . . 4 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) → (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 256	. . 3 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = 1_o) → dom 𝑇 ∈ dom 𝑇)
70	6, 69	mtand 814	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → ¬ (𝑈‘dom 𝑇) = 1_o)
71	70	neqned 2947	1 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → (𝑈‘dom 𝑇) ≠ 1_o)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {cab 2709 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ∪ cun 3945 ⊆ wss 3947 ∅c0 4321 ifcif 4527 {csn 4627 ⟨cop 4633 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5675 ↾ cres 5677 Ord word 6360 Oncon0 6361 suc csuc 6363 ℩cio 6490 ‘cfv 6540 ℩crio 7360 1_oc1o 8455 No csur 27132 <s cslt 27133

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7721

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6364 df-on 6365 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-1o 8462 df-2o 8463 df-no 27135 df-slt 27136 df-bday 27137

This theorem is referenced by: noinfbnd1lem4 27218 noinfbnd1lem5 27219 noinfbnd1lem6 27220

W3C validator