Theorem nosupbnd1lem3 27676

Description: Lemma for nosupbnd1 27680. 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 27669	. . . . 5 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
3	2	3ad2ant2 1134	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → 𝑆 ∈ No )
4		nodmord 27619	. . . 4 ⊢ (𝑆 ∈ No → Ord dom 𝑆)
5		ordirr 6333	. . . 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 1229	. . . . 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 6864	. . . . . . . 8 ⊢ (¬ dom 𝑆 ∈ dom 𝑈 → (𝑈‘dom 𝑆) = ∅)
9		2on 8408	. . . . . . . . . . . . 13 ⊢ 2o ∈ On
10	9	elexi 3461	. . . . . . . . . . . 12 ⊢ 2o ∈ V
11	10	prid2 4718	. . . . . . . . . . 11 ⊢ 2o ∈ {1o, 2o}
12	11	nosgnn0i 27625	. . . . . . . . . 10 ⊢ ∅ ≠ 2o
13		neeq1 2992	. . . . . . . . . 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 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 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 1227	. . . . . . . . . 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 3932	. . . . . . . 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 770	. . . . . . . . 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 3932	. . . . . . . 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 27616	. . . . . . . . 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 1230	. . . . . . . . . 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 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)) → ¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
32		simpll2 1214	. . . . . . . . . 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 1215	. . . . . . . . . 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 27675	. . . . . . . . . 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 1376	. . . . . . . . 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 2772	. . . . . . . 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 768	. . . . . . . 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 772	. . . . . . . 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 27638	. . . . . . . 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 1394	. . . . . . 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 3126	. . . . 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 5850	. . . . . . . 8 ⊢ (𝑝 = 𝑈 → dom 𝑝 = dom 𝑈)
45	44	eleq2d 2820	. . . . . . 7 ⊢ (𝑝 = 𝑈 → (dom 𝑆 ∈ dom 𝑝 ↔ dom 𝑆 ∈ dom 𝑈))
46		breq2 5100	. . . . . . . . . 10 ⊢ (𝑝 = 𝑈 → (𝑞 <s 𝑝 ↔ 𝑞 <s 𝑈))
47	46	notbid 318	. . . . . . . . 9 ⊢ (𝑝 = 𝑈 → (¬ 𝑞 <s 𝑝 ↔ ¬ 𝑞 <s 𝑈))
48		reseq1 5930	. . . . . . . . . 10 ⊢ (𝑝 = 𝑈 → (𝑝 ↾ suc dom 𝑆) = (𝑈 ↾ suc dom 𝑆))
49	48	eqeq1d 2736	. . . . . . . . 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 3157	. . . . . . 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 3574	. . . . 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 836	. . . 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 27670	. . . . . . . 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 2820	. . . . . . 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 1133	. . . . . 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 2822	. . . . . . . . . 10 ⊢ (𝑧 = dom 𝑆 → (𝑧 ∈ dom 𝑝 ↔ dom 𝑆 ∈ dom 𝑝))
59		suceq 6383	. . . . . . . . . . . . . 14 ⊢ (𝑧 = dom 𝑆 → suc 𝑧 = suc dom 𝑆)
60	59	reseq2d 5936	. . . . . . . . . . . . 13 ⊢ (𝑧 = dom 𝑆 → (𝑝 ↾ suc 𝑧) = (𝑝 ↾ suc dom 𝑆))
61	59	reseq2d 5936	. . . . . . . . . . . . 13 ⊢ (𝑧 = dom 𝑆 → (𝑞 ↾ suc 𝑧) = (𝑞 ↾ suc dom 𝑆))
62	60, 61	eqeq12d 2750	. . . . . . . . . . . 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 3157	. . . . . . . . . 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 3158	. . . . . . . 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 3629	. . . . . . 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 815	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  {cab 2712   ≠ wne 2930  ∀wral 3049  ∃wrex 3058  Vcvv 3438   ∪ cun 3897   ⊆ wss 3899  ∅c0 4283  ifcif 4477  {csn 4578  ⟨cop 4584   class class class wbr 5096   ↦ cmpt 5177  dom cdm 5622   ↾ cres 5624  Ord word 6314  Oncon0 6315  suc csuc 6317  ℩cio 6444  ‘cfv 6490  ℩crio 7312  1_oc1o 8388  2_oc2o 8389   No csur 27605   <s cslt 27606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4862  df-int 4901  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-ord 6318  df-on 6319  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fo 6496  df-fv 6498  df-riota 7313  df-1o 8395  df-2o 8396  df-no 27608  df-slt 27609  df-bday 27610

This theorem is referenced by:  nosupbnd1lem4  27677  nosupbnd1lem5  27678  nosupbnd1lem6  27679