Theorem nosupbnd1lem4 33213

Description: Lemma for nosupbnd1 33216. If 𝑈 is a prolongment of 𝑆 and in 𝐴, then (𝑈‘dom 𝑆) is not undefined. (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
nosupbnd1lem4	⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ ∅)

Distinct variable groups:   𝐴,𝑔,𝑢,𝑣,𝑥,𝑦   𝑢,𝑈   𝑣,𝑢,𝑥,𝑦

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

Proof of Theorem nosupbnd1lem4

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1187	. . . . . . . 8 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → ¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
2		simpl2 1188	. . . . . . . 8 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → (𝐴 ⊆ No ∧ 𝐴 ∈ V))
3		simprl 769	. . . . . . . 8 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → 𝑤 ∈ 𝐴)
4		simpl3 1189	. . . . . . . . 9 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆))
5		simprr 771	. . . . . . . . . . 11 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → 𝑈 <s 𝑤)
6		simp2l 1195	. . . . . . . . . . . . 13 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → 𝐴 ⊆ No )
7		simp3l 1197	. . . . . . . . . . . . 13 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → 𝑈 ∈ 𝐴)
8	6, 7	sseldd 3970	. . . . . . . . . . . 12 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → 𝑈 ∈ No )
9		simpl2l 1222	. . . . . . . . . . . . 13 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → 𝐴 ⊆ No )
10	9, 3	sseldd 3970	. . . . . . . . . . . 12 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → 𝑤 ∈ No )
11		sltso 33183	. . . . . . . . . . . . 13 ⊢ <s Or No
12		soasym 5506	. . . . . . . . . . . . 13 ⊢ (( <s Or No ∧ (𝑈 ∈ No ∧ 𝑤 ∈ No )) → (𝑈 <s 𝑤 → ¬ 𝑤 <s 𝑈))
13	11, 12	mpan 688	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ No ∧ 𝑤 ∈ No ) → (𝑈 <s 𝑤 → ¬ 𝑤 <s 𝑈))
14	8, 10, 13	syl2an2r 683	. . . . . . . . . . 11 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → (𝑈 <s 𝑤 → ¬ 𝑤 <s 𝑈))
15	5, 14	mpd 15	. . . . . . . . . 10 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → ¬ 𝑤 <s 𝑈)
16	3, 15	jca 514	. . . . . . . . 9 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → (𝑤 ∈ 𝐴 ∧ ¬ 𝑤 <s 𝑈))
17		nosupbnd1.1	. . . . . . . . . 10 ⊢ 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))))
18	17	nosupbnd1lem2 33211	. . . . . . . . 9 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ ((𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑤 ∈ 𝐴 ∧ ¬ 𝑤 <s 𝑈))) → (𝑤 ↾ dom 𝑆) = 𝑆)
19	1, 2, 4, 16, 18	syl112anc 1370	. . . . . . . 8 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → (𝑤 ↾ dom 𝑆) = 𝑆)
20	17	nosupbnd1lem3 33212	. . . . . . . 8 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑤 ∈ 𝐴 ∧ (𝑤 ↾ dom 𝑆) = 𝑆)) → (𝑤‘dom 𝑆) ≠ 2o)
21	1, 2, 3, 19, 20	syl112anc 1370	. . . . . . 7 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → (𝑤‘dom 𝑆) ≠ 2o)
22	21	neneqd 3023	. . . . . 6 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → ¬ (𝑤‘dom 𝑆) = 2o)
23	22	expr 459	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ 𝑤 ∈ 𝐴) → (𝑈 <s 𝑤 → ¬ (𝑤‘dom 𝑆) = 2o))
24		imnan 402	. . . . 5 ⊢ ((𝑈 <s 𝑤 → ¬ (𝑤‘dom 𝑆) = 2o) ↔ ¬ (𝑈 <s 𝑤 ∧ (𝑤‘dom 𝑆) = 2o))
25	23, 24	sylib 220	. . . 4 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ 𝑤 ∈ 𝐴) → ¬ (𝑈 <s 𝑤 ∧ (𝑤‘dom 𝑆) = 2o))
26	25	nrexdv 3272	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → ¬ ∃𝑤 ∈ 𝐴 (𝑈 <s 𝑤 ∧ (𝑤‘dom 𝑆) = 2o))
27		simpl3l 1224	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) → 𝑈 ∈ 𝐴)
28		simpl1 1187	. . . . . 6 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) → ¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
29		breq2 5072	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑢 <s 𝑤 ↔ 𝑢 <s 𝑦))
30	29	cbvrexvw 3452	. . . . . . . . 9 ⊢ (∃𝑤 ∈ 𝐴 𝑢 <s 𝑤 ↔ ∃𝑦 ∈ 𝐴 𝑢 <s 𝑦)
31		breq1 5071	. . . . . . . . . 10 ⊢ (𝑢 = 𝑥 → (𝑢 <s 𝑦 ↔ 𝑥 <s 𝑦))
32	31	rexbidv 3299	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → (∃𝑦 ∈ 𝐴 𝑢 <s 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑥 <s 𝑦))
33	30, 32	syl5bb 285	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → (∃𝑤 ∈ 𝐴 𝑢 <s 𝑤 ↔ ∃𝑦 ∈ 𝐴 𝑥 <s 𝑦))
34	33	cbvralvw 3451	. . . . . . 7 ⊢ (∀𝑢 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑢 <s 𝑤 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <s 𝑦)
35		dfrex2 3241	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐴 𝑥 <s 𝑦 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
36	35	ralbii 3167	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <s 𝑦 ↔ ∀𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
37		ralnex 3238	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ↔ ¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
38	34, 36, 37	3bitri 299	. . . . . 6 ⊢ (∀𝑢 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑢 <s 𝑤 ↔ ¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
39	28, 38	sylibr 236	. . . . 5 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) → ∀𝑢 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑢 <s 𝑤)
40		breq1 5071	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (𝑢 <s 𝑤 ↔ 𝑈 <s 𝑤))
41	40	rexbidv 3299	. . . . . 6 ⊢ (𝑢 = 𝑈 → (∃𝑤 ∈ 𝐴 𝑢 <s 𝑤 ↔ ∃𝑤 ∈ 𝐴 𝑈 <s 𝑤))
42	41	rspcv 3620	. . . . 5 ⊢ (𝑈 ∈ 𝐴 → (∀𝑢 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑢 <s 𝑤 → ∃𝑤 ∈ 𝐴 𝑈 <s 𝑤))
43	27, 39, 42	sylc 65	. . . 4 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) → ∃𝑤 ∈ 𝐴 𝑈 <s 𝑤)
44		simpl2l 1222	. . . . . . . . . 10 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) → 𝐴 ⊆ No )
45	44, 27	sseldd 3970	. . . . . . . . 9 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) → 𝑈 ∈ No )
46	45	adantr 483	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → 𝑈 ∈ No )
47	44	adantr 483	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → 𝐴 ⊆ No )
48		simprl 769	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → 𝑤 ∈ 𝐴)
49	47, 48	sseldd 3970	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → 𝑤 ∈ No )
50	17	nosupno 33205	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
51	50	3ad2ant2 1130	. . . . . . . . . . 11 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → 𝑆 ∈ No )
52	51	adantr 483	. . . . . . . . . 10 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) → 𝑆 ∈ No )
53	52	adantr 483	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → 𝑆 ∈ No )
54		nodmon 33159	. . . . . . . . 9 ⊢ (𝑆 ∈ No → dom 𝑆 ∈ On)
55	53, 54	syl 17	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → dom 𝑆 ∈ On)
56		simpl3r 1225	. . . . . . . . . 10 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) → (𝑈 ↾ dom 𝑆) = 𝑆)
57	56	adantr 483	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → (𝑈 ↾ dom 𝑆) = 𝑆)
58		simpll1 1208	. . . . . . . . . 10 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → ¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
59		simpll2 1209	. . . . . . . . . 10 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → (𝐴 ⊆ No ∧ 𝐴 ∈ V))
60		simpll3 1210	. . . . . . . . . 10 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆))
61		simprr 771	. . . . . . . . . . . 12 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → 𝑈 <s 𝑤)
62	45, 49, 13	syl2an2r 683	. . . . . . . . . . . 12 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → (𝑈 <s 𝑤 → ¬ 𝑤 <s 𝑈))
63	61, 62	mpd 15	. . . . . . . . . . 11 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → ¬ 𝑤 <s 𝑈)
64	48, 63	jca 514	. . . . . . . . . 10 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → (𝑤 ∈ 𝐴 ∧ ¬ 𝑤 <s 𝑈))
65	58, 59, 60, 64, 18	syl112anc 1370	. . . . . . . . 9 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → (𝑤 ↾ dom 𝑆) = 𝑆)
66	57, 65	eqtr4d 2861	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → (𝑈 ↾ dom 𝑆) = (𝑤 ↾ dom 𝑆))
67		simplr 767	. . . . . . . 8 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → (𝑈‘dom 𝑆) = ∅)
68		nolt02o 33201	. . . . . . . 8 ⊢ (((𝑈 ∈ No ∧ 𝑤 ∈ No ∧ dom 𝑆 ∈ On) ∧ ((𝑈 ↾ dom 𝑆) = (𝑤 ↾ dom 𝑆) ∧ 𝑈 <s 𝑤) ∧ (𝑈‘dom 𝑆) = ∅) → (𝑤‘dom 𝑆) = 2o)
69	46, 49, 55, 66, 61, 67, 68	syl321anc 1388	. . . . . . 7 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) ∧ (𝑤 ∈ 𝐴 ∧ 𝑈 <s 𝑤)) → (𝑤‘dom 𝑆) = 2o)
70	69	expr 459	. . . . . 6 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) ∧ 𝑤 ∈ 𝐴) → (𝑈 <s 𝑤 → (𝑤‘dom 𝑆) = 2o))
71	70	ancld 553	. . . . 5 ⊢ ((((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) ∧ 𝑤 ∈ 𝐴) → (𝑈 <s 𝑤 → (𝑈 <s 𝑤 ∧ (𝑤‘dom 𝑆) = 2o)))
72	71	reximdva 3276	. . . 4 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) → (∃𝑤 ∈ 𝐴 𝑈 <s 𝑤 → ∃𝑤 ∈ 𝐴 (𝑈 <s 𝑤 ∧ (𝑤‘dom 𝑆) = 2o)))
73	43, 72	mpd 15	. . 3 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) ∧ (𝑈‘dom 𝑆) = ∅) → ∃𝑤 ∈ 𝐴 (𝑈 <s 𝑤 ∧ (𝑤‘dom 𝑆) = 2o))
74	26, 73	mtand 814	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → ¬ (𝑈‘dom 𝑆) = ∅)
75	74	neqned 3025	1 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  {cab 2801   ≠ wne 3018  ∀wral 3140  ∃wrex 3141  Vcvv 3496   ∪ cun 3936   ⊆ wss 3938  ∅c0 4293  ifcif 4469  {csn 4569  ⟨cop 4575   class class class wbr 5068   ↦ cmpt 5148   Or wor 5475  dom cdm 5557   ↾ cres 5559  Oncon0 6193  suc csuc 6195  ℩cio 6314  ‘cfv 6357  ℩crio 7115  2_oc2o 8098   No csur 33149   <s cslt 33150

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-ord 6196  df-on 6197  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-1o 8104  df-2o 8105  df-no 33152  df-slt 33153  df-bday 33154

This theorem is referenced by:  nosupbnd1lem5  33214  nosupbnd1lem6  33215