Theorem nosupbnd1lem3 27770

Description: Lemma for nosupbnd1 27774. 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 27763	. . . . 5 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
3	2	3ad2ant2 1133	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → 𝑆 ∈ No )
4		nodmord 27713	. . . 4 ⊢ (𝑆 ∈ No → Ord dom 𝑆)
5		ordirr 6404	. . . 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 1227	. . . . 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 6942	. . . . . . . 8 ⊢ (¬ dom 𝑆 ∈ dom 𝑈 → (𝑈‘dom 𝑆) = ∅)
9		2on 8519	. . . . . . . . . . . . 13 ⊢ 2o ∈ On
10	9	elexi 3501	. . . . . . . . . . . 12 ⊢ 2o ∈ V
11	10	prid2 4768	. . . . . . . . . . 11 ⊢ 2o ∈ {1o, 2o}
12	11	nosgnn0i 27719	. . . . . . . . . 10 ⊢ ∅ ≠ 2o
13		neeq1 3001	. . . . . . . . . 10 ⊢ ((𝑈‘dom 𝑆) = ∅ → ((𝑈‘dom 𝑆) ≠ 2o ↔ ∅ ≠ 2o))
14	12, 13	mpbiri 258	. . . . . . . . 9 ⊢ ((𝑈‘dom 𝑆) = ∅ → (𝑈‘dom 𝑆) ≠ 2o)
15	14	neneqd 2943	. . . . . . . 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 481	. . . . 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 1225	. . . . . . . . . 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 480	. . . . . . . . 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 480	. . . . . . . . 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 3996	. . . . . . . 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 771	. . . . . . . . 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 3996	. . . . . . . 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 480	. . . . . . . . . 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 480	. . . . . . . . 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 27710	. . . . . . . . 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 1228	. . . . . . . . . 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 480	. . . . . . . . 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 1211	. . . . . . . . . 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 1212	. . . . . . . . . 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 1213	. . . . . . . . . 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 484	. . . . . . . . . 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 27769	. . . . . . . . . 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 1373	. . . . . . . . 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 2778	. . . . . . . 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 769	. . . . . . . 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 773	. . . . . . . 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 27732	. . . . . . . 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 1391	. . . . . . 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 456	. . . . . 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 3144	. . . . 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 5917	. . . . . . . 8 ⊢ (𝑝 = 𝑈 → dom 𝑝 = dom 𝑈)
45	44	eleq2d 2825	. . . . . . 7 ⊢ (𝑝 = 𝑈 → (dom 𝑆 ∈ dom 𝑝 ↔ dom 𝑆 ∈ dom 𝑈))
46		breq2 5152	. . . . . . . . . 10 ⊢ (𝑝 = 𝑈 → (𝑞 <s 𝑝 ↔ 𝑞 <s 𝑈))
47	46	notbid 318	. . . . . . . . 9 ⊢ (𝑝 = 𝑈 → (¬ 𝑞 <s 𝑝 ↔ ¬ 𝑞 <s 𝑈))
48		reseq1 5994	. . . . . . . . . 10 ⊢ (𝑝 = 𝑈 → (𝑝 ↾ suc dom 𝑆) = (𝑈 ↾ suc dom 𝑆))
49	48	eqeq1d 2737	. . . . . . . . 9 ⊢ (𝑝 = 𝑈 → ((𝑝 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆) ↔ (𝑈 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆)))
50	47, 49	imbi12d 344	. . . . . . . 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 3176	. . . . . . 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 632	. . . . . 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 3622	. . . . 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 837	. . . 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 27764	. . . . . . . 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 2825	. . . . . . 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 1132	. . . . . 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 2827	. . . . . . . . . 10 ⊢ (𝑧 = dom 𝑆 → (𝑧 ∈ dom 𝑝 ↔ dom 𝑆 ∈ dom 𝑝))
59		suceq 6452	. . . . . . . . . . . . . 14 ⊢ (𝑧 = dom 𝑆 → suc 𝑧 = suc dom 𝑆)
60	59	reseq2d 6000	. . . . . . . . . . . . 13 ⊢ (𝑧 = dom 𝑆 → (𝑝 ↾ suc 𝑧) = (𝑝 ↾ suc dom 𝑆))
61	59	reseq2d 6000	. . . . . . . . . . . . 13 ⊢ (𝑧 = dom 𝑆 → (𝑞 ↾ suc 𝑧) = (𝑞 ↾ suc dom 𝑆))
62	60, 61	eqeq12d 2751	. . . . . . . . . . . 12 ⊢ (𝑧 = dom 𝑆 → ((𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧) ↔ (𝑝 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆)))
63	62	imbi2d 340	. . . . . . . . . . 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 3176	. . . . . . . . . 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 632	. . . . . . . . 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 3177	. . . . . . . 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 3677	. . . . . . 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 279	. . . . 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 480	. . . 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 257	. . 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 816	. 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 2945	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  {cab 2712   ≠ wne 2938  ∀wral 3059  ∃wrex 3068  Vcvv 3478   ∪ cun 3961   ⊆ wss 3963  ∅c0 4339  ifcif 4531  {csn 4631  ⟨cop 4637   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5689   ↾ cres 5691  Ord word 6385  Oncon0 6386  suc csuc 6388  ℩cio 6514  ‘cfv 6563  ℩crio 7387  1_oc1o 8498  2_oc2o 8499   No csur 27699   <s cslt 27700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-riota 7388  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704

This theorem is referenced by:  nosupbnd1lem4  27771  nosupbnd1lem5  27772  nosupbnd1lem6  27773