Theorem nosupbnd1lem3 32300

Description: Lemma for nosupbnd1 32304. If 𝑈 is a prolongment of 𝑆 and in 𝐴, then (𝑈‘dom 𝑆) is not 2_𝑜. (Contributed by Scott Fenton, 6-Dec-2021.)

Hypothesis

Ref	Expression
nosupbnd1.1	⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (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))))

Assertion

Ref	Expression
nosupbnd1lem3	⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ 2𝑜)

Distinct variable group:   𝐴,𝑔,𝑢,𝑣,𝑥,𝑦

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

Proof of Theorem nosupbnd1lem3

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

Step	Hyp	Ref	Expression
1		nosupbnd1.1	. . . . . 6 ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (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))))
2	1	nosupno 32293	. . . . 5 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
3	2	3ad2ant2 1164	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → 𝑆 ∈ No )
4		nodmord 32250	. . . 4 ⊢ (𝑆 ∈ No → Ord dom 𝑆)
5		ordirr 5926	. . . 4 ⊢ (Ord dom 𝑆 → ¬ dom 𝑆 ∈ dom 𝑆)
6	3, 4, 5	3syl 18	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → ¬ dom 𝑆 ∈ dom 𝑆)
7		simpl3l 1301	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) → 𝑈 ∈ 𝐴)
8		ndmfv 6405	. . . . . . . 8 ⊢ (¬ dom 𝑆 ∈ dom 𝑈 → (𝑈‘dom 𝑆) = ∅)
9		2on 7773	. . . . . . . . . . . . 13 ⊢ 2𝑜 ∈ On
10	9	elexi 3366	. . . . . . . . . . . 12 ⊢ 2𝑜 ∈ V
11	10	prid2 4453	. . . . . . . . . . 11 ⊢ 2𝑜 ∈ {1𝑜, 2𝑜}
12	11	nosgnn0i 32256	. . . . . . . . . 10 ⊢ ∅ ≠ 2𝑜
13		neeq1 2999	. . . . . . . . . 10 ⊢ ((𝑈‘dom 𝑆) = ∅ → ((𝑈‘dom 𝑆) ≠ 2𝑜 ↔ ∅ ≠ 2𝑜))
14	12, 13	mpbiri 249	. . . . . . . . 9 ⊢ ((𝑈‘dom 𝑆) = ∅ → (𝑈‘dom 𝑆) ≠ 2𝑜)
15	14	neneqd 2942	. . . . . . . 8 ⊢ ((𝑈‘dom 𝑆) = ∅ → ¬ (𝑈‘dom 𝑆) = 2𝑜)
16	8, 15	syl 17	. . . . . . 7 ⊢ (¬ dom 𝑆 ∈ dom 𝑈 → ¬ (𝑈‘dom 𝑆) = 2𝑜)
17	16	con4i 114	. . . . . 6 ⊢ ((𝑈‘dom 𝑆) = 2𝑜 → dom 𝑆 ∈ dom 𝑈)
18	17	adantl 473	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) → dom 𝑆 ∈ dom 𝑈)
19		simpl2l 1297	. . . . . . . . . 10 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) → 𝐴 ⊆ No )
20	19	adantr 472	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 <s 𝑈)) → 𝐴 ⊆ No )
21	7	adantr 472	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 <s 𝑈)) → 𝑈 ∈ 𝐴)
22	20, 21	sseldd 3762	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 <s 𝑈)) → 𝑈 ∈ No )
23		simprl 787	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 <s 𝑈)) → 𝑞 ∈ 𝐴)
24	20, 23	sseldd 3762	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 <s 𝑈)) → 𝑞 ∈ No )
25	3	adantr 472	. . . . . . . . . 10 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) → 𝑆 ∈ No )
26	25	adantr 472	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 <s 𝑈)) → 𝑆 ∈ No )
27		nodmon 32247	. . . . . . . . 9 ⊢ (𝑆 ∈ No → dom 𝑆 ∈ On)
28	26, 27	syl 17	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 <s 𝑈)) → dom 𝑆 ∈ On)
29		simpl3r 1303	. . . . . . . . . 10 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) → (𝑈 ↾ dom 𝑆) = 𝑆)
30	29	adantr 472	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 <s 𝑈)) → (𝑈 ↾ dom 𝑆) = 𝑆)
31		simpll1 1269	. . . . . . . . . 10 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 <s 𝑈)) → ¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
32		simpll2 1271	. . . . . . . . . 10 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 <s 𝑈)) → (𝐴 ⊆ No ∧ 𝐴 ∈ V))
33		simpll3 1273	. . . . . . . . . 10 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 <s 𝑈)) → (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆))
34		simpr 477	. . . . . . . . . 10 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 <s 𝑈)) → (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 <s 𝑈))
35	1	nosupbnd1lem2 32299	. . . . . . . . . 10 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ ((𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 <s 𝑈))) → (𝑞 ↾ dom 𝑆) = 𝑆)
36	31, 32, 33, 34, 35	syl112anc 1493	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 <s 𝑈)) → (𝑞 ↾ dom 𝑆) = 𝑆)
37	30, 36	eqtr4d 2802	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 <s 𝑈)) → (𝑈 ↾ dom 𝑆) = (𝑞 ↾ dom 𝑆))
38		simplr 785	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 <s 𝑈)) → (𝑈‘dom 𝑆) = 2𝑜)
39		simprr 789	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 <s 𝑈)) → ¬ 𝑞 <s 𝑈)
40		nolesgn2ores 32269	. . . . . . . 8 ⊢ (((𝑈 ∈ No ∧ 𝑞 ∈ No ∧ dom 𝑆 ∈ On) ∧ ((𝑈 ↾ dom 𝑆) = (𝑞 ↾ dom 𝑆) ∧ (𝑈‘dom 𝑆) = 2𝑜) ∧ ¬ 𝑞 <s 𝑈) → (𝑈 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆))
41	22, 24, 28, 37, 38, 39, 40	syl321anc 1511	. . . . . . 7 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 <s 𝑈)) → (𝑈 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆))
42	41	expr 448	. . . . . 6 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) ∧ 𝑞 ∈ 𝐴) → (¬ 𝑞 <s 𝑈 → (𝑈 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆)))
43	42	ralrimiva 3113	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) → ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑈 → (𝑈 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆)))
44		dmeq 5492	. . . . . . . 8 ⊢ (𝑝 = 𝑈 → dom 𝑝 = dom 𝑈)
45	44	eleq2d 2830	. . . . . . 7 ⊢ (𝑝 = 𝑈 → (dom 𝑆 ∈ dom 𝑝 ↔ dom 𝑆 ∈ dom 𝑈))
46		breq2 4813	. . . . . . . . . 10 ⊢ (𝑝 = 𝑈 → (𝑞 <s 𝑝 ↔ 𝑞 <s 𝑈))
47	46	notbid 309	. . . . . . . . 9 ⊢ (𝑝 = 𝑈 → (¬ 𝑞 <s 𝑝 ↔ ¬ 𝑞 <s 𝑈))
48		reseq1 5559	. . . . . . . . . 10 ⊢ (𝑝 = 𝑈 → (𝑝 ↾ suc dom 𝑆) = (𝑈 ↾ suc dom 𝑆))
49	48	eqeq1d 2767	. . . . . . . . 9 ⊢ (𝑝 = 𝑈 → ((𝑝 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆) ↔ (𝑈 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆)))
50	47, 49	imbi12d 335	. . . . . . . 8 ⊢ (𝑝 = 𝑈 → ((¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆)) ↔ (¬ 𝑞 <s 𝑈 → (𝑈 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆))))
51	50	ralbidv 3133	. . . . . . 7 ⊢ (𝑝 = 𝑈 → (∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆)) ↔ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑈 → (𝑈 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆))))
52	45, 51	anbi12d 624	. . . . . 6 ⊢ (𝑝 = 𝑈 → ((dom 𝑆 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆))) ↔ (dom 𝑆 ∈ dom 𝑈 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑈 → (𝑈 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆)))))
53	52	rspcev 3461	. . . . 5 ⊢ ((𝑈 ∈ 𝐴 ∧ (dom 𝑆 ∈ dom 𝑈 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑈 → (𝑈 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆)))) → ∃𝑝 ∈ 𝐴 (dom 𝑆 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆))))
54	7, 18, 43, 53	syl12anc 865	. . . 4 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) → ∃𝑝 ∈ 𝐴 (dom 𝑆 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆))))
55	1	nosupdm 32294	. . . . . . . 8 ⊢ (¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = {𝑧 ∣ ∃𝑝 ∈ 𝐴 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))})
56	55	eleq2d 2830	. . . . . . 7 ⊢ (¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → (dom 𝑆 ∈ dom 𝑆 ↔ dom 𝑆 ∈ {𝑧 ∣ ∃𝑝 ∈ 𝐴 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))}))
57	56	3ad2ant1 1163	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (dom 𝑆 ∈ dom 𝑆 ↔ dom 𝑆 ∈ {𝑧 ∣ ∃𝑝 ∈ 𝐴 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))}))
58		eleq1 2832	. . . . . . . . . 10 ⊢ (𝑧 = dom 𝑆 → (𝑧 ∈ dom 𝑝 ↔ dom 𝑆 ∈ dom 𝑝))
59		suceq 5973	. . . . . . . . . . . . . 14 ⊢ (𝑧 = dom 𝑆 → suc 𝑧 = suc dom 𝑆)
60	59	reseq2d 5565	. . . . . . . . . . . . 13 ⊢ (𝑧 = dom 𝑆 → (𝑝 ↾ suc 𝑧) = (𝑝 ↾ suc dom 𝑆))
61	59	reseq2d 5565	. . . . . . . . . . . . 13 ⊢ (𝑧 = dom 𝑆 → (𝑞 ↾ suc 𝑧) = (𝑞 ↾ suc dom 𝑆))
62	60, 61	eqeq12d 2780	. . . . . . . . . . . 12 ⊢ (𝑧 = dom 𝑆 → ((𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧) ↔ (𝑝 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆)))
63	62	imbi2d 331	. . . . . . . . . . 11 ⊢ (𝑧 = dom 𝑆 → ((¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)) ↔ (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆))))
64	63	ralbidv 3133	. . . . . . . . . 10 ⊢ (𝑧 = dom 𝑆 → (∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)) ↔ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆))))
65	58, 64	anbi12d 624	. . . . . . . . 9 ⊢ (𝑧 = dom 𝑆 → ((𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧))) ↔ (dom 𝑆 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆)))))
66	65	rexbidv 3199	. . . . . . . 8 ⊢ (𝑧 = dom 𝑆 → (∃𝑝 ∈ 𝐴 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧))) ↔ ∃𝑝 ∈ 𝐴 (dom 𝑆 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆)))))
67	66	elabg 3505	. . . . . . 7 ⊢ (dom 𝑆 ∈ On → (dom 𝑆 ∈ {𝑧 ∣ ∃𝑝 ∈ 𝐴 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))} ↔ ∃𝑝 ∈ 𝐴 (dom 𝑆 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆)))))
68	3, 27, 67	3syl 18	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (dom 𝑆 ∈ {𝑧 ∣ ∃𝑝 ∈ 𝐴 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))} ↔ ∃𝑝 ∈ 𝐴 (dom 𝑆 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆)))))
69	57, 68	bitrd 270	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (dom 𝑆 ∈ dom 𝑆 ↔ ∃𝑝 ∈ 𝐴 (dom 𝑆 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆)))))
70	69	adantr 472	. . . 4 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) → (dom 𝑆 ∈ dom 𝑆 ↔ ∃𝑝 ∈ 𝐴 (dom 𝑆 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆)))))
71	54, 70	mpbird 248	. . 3 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = 2𝑜) → dom 𝑆 ∈ dom 𝑆)
72	6, 71	mtand 850	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → ¬ (𝑈‘dom 𝑆) = 2𝑜)
73	72	neqned 2944	1 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ 2𝑜)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 197   ∧ wa 384   ∧ w3a 1107   = wceq 1652   ∈ wcel 2155  {cab 2751   ≠ wne 2937  ∀wral 3055  ∃wrex 3056  Vcvv 3350   ∪ cun 3730   ⊆ wss 3732  ∅c0 4079  ifcif 4243  {csn 4334  ⟨cop 4340   class class class wbr 4809   ↦ cmpt 4888  dom cdm 5277   ↾ cres 5279  Ord word 5907  Oncon0 5908  suc csuc 5910  ℩cio 6029  ‘cfv 6068  ℩crio 6802  1_𝑜c1o 7757  2_𝑜c2o 7758   No csur 32237   <s cslt 32238

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-1o 7764  df-2o 7765  df-no 32240  df-slt 32241  df-bday 32242

This theorem is referenced by:  nosupbnd1lem4  32301  nosupbnd1lem5  32302  nosupbnd1lem6  32303