Theorem nosupbnd1lem3 27656

Description: Lemma for nosupbnd1 27660. 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 27649	. . . . 5 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
3	2	3ad2ant2 1131	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → 𝑆 ∈ No )
4		nodmord 27599	. . . 4 ⊢ (𝑆 ∈ No → Ord dom 𝑆)
5		ordirr 6383	. . . 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 1225	. . . . 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 6925	. . . . . . . 8 ⊢ (¬ dom 𝑆 ∈ dom 𝑈 → (𝑈‘dom 𝑆) = ∅)
9		2on 8494	. . . . . . . . . . . . 13 ⊢ 2o ∈ On
10	9	elexi 3484	. . . . . . . . . . . 12 ⊢ 2o ∈ V
11	10	prid2 4764	. . . . . . . . . . 11 ⊢ 2o ∈ {1o, 2o}
12	11	nosgnn0i 27605	. . . . . . . . . 10 ⊢ ∅ ≠ 2o
13		neeq1 2993	. . . . . . . . . 10 ⊢ ((𝑈‘dom 𝑆) = ∅ → ((𝑈‘dom 𝑆) ≠ 2o ↔ ∅ ≠ 2o))
14	12, 13	mpbiri 257	. . . . . . . . 9 ⊢ ((𝑈‘dom 𝑆) = ∅ → (𝑈‘dom 𝑆) ≠ 2o)
15	14	neneqd 2935	. . . . . . . 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 480	. . . . 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 1223	. . . . . . . . . 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 479	. . . . . . . . 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 479	. . . . . . . . 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 3974	. . . . . . . 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 769	. . . . . . . . 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 3974	. . . . . . . 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 479	. . . . . . . . . 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 479	. . . . . . . . 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 27596	. . . . . . . . 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 1226	. . . . . . . . . 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 479	. . . . . . . . 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 1209	. . . . . . . . . 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 1210	. . . . . . . . . 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 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)) → (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆))
34		simpr 483	. . . . . . . . . 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 27655	. . . . . . . . . 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 1371	. . . . . . . . 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 2768	. . . . . . . 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 767	. . . . . . . 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 771	. . . . . . . 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 27618	. . . . . . . 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 1389	. . . . . . 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 455	. . . . . 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 3136	. . . . 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 5901	. . . . . . . 8 ⊢ (𝑝 = 𝑈 → dom 𝑝 = dom 𝑈)
45	44	eleq2d 2811	. . . . . . 7 ⊢ (𝑝 = 𝑈 → (dom 𝑆 ∈ dom 𝑝 ↔ dom 𝑆 ∈ dom 𝑈))
46		breq2 5148	. . . . . . . . . 10 ⊢ (𝑝 = 𝑈 → (𝑞 <s 𝑝 ↔ 𝑞 <s 𝑈))
47	46	notbid 317	. . . . . . . . 9 ⊢ (𝑝 = 𝑈 → (¬ 𝑞 <s 𝑝 ↔ ¬ 𝑞 <s 𝑈))
48		reseq1 5974	. . . . . . . . . 10 ⊢ (𝑝 = 𝑈 → (𝑝 ↾ suc dom 𝑆) = (𝑈 ↾ suc dom 𝑆))
49	48	eqeq1d 2727	. . . . . . . . 9 ⊢ (𝑝 = 𝑈 → ((𝑝 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆) ↔ (𝑈 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆)))
50	47, 49	imbi12d 343	. . . . . . . 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 3168	. . . . . . 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 630	. . . . . 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 3603	. . . . 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 835	. . . 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 27650	. . . . . . . 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 2811	. . . . . . 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 1130	. . . . . 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 2813	. . . . . . . . . 10 ⊢ (𝑧 = dom 𝑆 → (𝑧 ∈ dom 𝑝 ↔ dom 𝑆 ∈ dom 𝑝))
59		suceq 6431	. . . . . . . . . . . . . 14 ⊢ (𝑧 = dom 𝑆 → suc 𝑧 = suc dom 𝑆)
60	59	reseq2d 5980	. . . . . . . . . . . . 13 ⊢ (𝑧 = dom 𝑆 → (𝑝 ↾ suc 𝑧) = (𝑝 ↾ suc dom 𝑆))
61	59	reseq2d 5980	. . . . . . . . . . . . 13 ⊢ (𝑧 = dom 𝑆 → (𝑞 ↾ suc 𝑧) = (𝑞 ↾ suc dom 𝑆))
62	60, 61	eqeq12d 2741	. . . . . . . . . . . 12 ⊢ (𝑧 = dom 𝑆 → ((𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧) ↔ (𝑝 ↾ suc dom 𝑆) = (𝑞 ↾ suc dom 𝑆)))
63	62	imbi2d 339	. . . . . . . . . . 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 3168	. . . . . . . . . 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 630	. . . . . . . . 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 3169	. . . . . . . 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 3659	. . . . . . 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 278	. . . . 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 479	. . . 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 256	. . 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 814	. 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 2937	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 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2702   ≠ wne 2930  ∀wral 3051  ∃wrex 3060  Vcvv 3463   ∪ cun 3939   ⊆ wss 3941  ∅c0 4319  ifcif 4525  {csn 4625  ⟨cop 4631   class class class wbr 5144   ↦ cmpt 5227  dom cdm 5673   ↾ cres 5675  Ord word 6364  Oncon0 6365  suc csuc 6367  ℩cio 6493  ‘cfv 6543  ℩crio 7368  1_oc1o 8473  2_oc2o 8474   No csur 27586   <s cslt 27587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pr 5424  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-1o 8480  df-2o 8481  df-no 27589  df-slt 27590  df-bday 27591

This theorem is referenced by:  nosupbnd1lem4  27657  nosupbnd1lem5  27658  nosupbnd1lem6  27659