Theorem nosupbnd1lem4 27599

Description: Lemma for nosupbnd1 27602. 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 1188	. . . . . . . 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 1189	. . . . . . . 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 768	. . . . . . . 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 1190	. . . . . . . . 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 770	. . . . . . . . . . 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 1196	. . . . . . . . . . . . 13 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → 𝐴 ⊆ No )
7		simp3l 1198	. . . . . . . . . . . . 13 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → 𝑈 ∈ 𝐴)
8	6, 7	sseldd 3978	. . . . . . . . . . . 12 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → 𝑈 ∈ No )
9		simpl2l 1223	. . . . . . . . . . . . 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 3978	. . . . . . . . . . . 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 27564	. . . . . . . . . . . . 13 ⊢ <s Or No
12		soasym 5612	. . . . . . . . . . . . 13 ⊢ (( <s Or No ∧ (𝑈 ∈ No ∧ 𝑤 ∈ No )) → (𝑈 <s 𝑤 → ¬ 𝑤 <s 𝑈))
13	11, 12	mpan 687	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ No ∧ 𝑤 ∈ No ) → (𝑈 <s 𝑤 → ¬ 𝑤 <s 𝑈))
14	8, 10, 13	syl2an2r 682	. . . . . . . . . . 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 511	. . . . . . . . 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 27597	. . . . . . . . 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 1371	. . . . . . . 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 27598	. . . . . . . 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 1371	. . . . . . 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 2939	. . . . . 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 456	. . . . 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 399	. . . . 5 ⊢ ((𝑈 <s 𝑤 → ¬ (𝑤‘dom 𝑆) = 2o) ↔ ¬ (𝑈 <s 𝑤 ∧ (𝑤‘dom 𝑆) = 2o))
25	23, 24	sylib 217	. . . 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 3143	. . 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 1225	. . . . 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 1188	. . . . . 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 5145	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑢 <s 𝑤 ↔ 𝑢 <s 𝑦))
30	29	cbvrexvw 3229	. . . . . . . . 9 ⊢ (∃𝑤 ∈ 𝐴 𝑢 <s 𝑤 ↔ ∃𝑦 ∈ 𝐴 𝑢 <s 𝑦)
31		breq1 5144	. . . . . . . . . 10 ⊢ (𝑢 = 𝑥 → (𝑢 <s 𝑦 ↔ 𝑥 <s 𝑦))
32	31	rexbidv 3172	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → (∃𝑦 ∈ 𝐴 𝑢 <s 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑥 <s 𝑦))
33	30, 32	bitrid 283	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → (∃𝑤 ∈ 𝐴 𝑢 <s 𝑤 ↔ ∃𝑦 ∈ 𝐴 𝑥 <s 𝑦))
34	33	cbvralvw 3228	. . . . . . 7 ⊢ (∀𝑢 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑢 <s 𝑤 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <s 𝑦)
35		dfrex2 3067	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐴 𝑥 <s 𝑦 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
36	35	ralbii 3087	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 <s 𝑦 ↔ ∀𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
37		ralnex 3066	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ↔ ¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
38	34, 36, 37	3bitri 297	. . . . . 6 ⊢ (∀𝑢 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑢 <s 𝑤 ↔ ¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
39	28, 38	sylibr 233	. . . . 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 5144	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (𝑢 <s 𝑤 ↔ 𝑈 <s 𝑤))
41	40	rexbidv 3172	. . . . . 6 ⊢ (𝑢 = 𝑈 → (∃𝑤 ∈ 𝐴 𝑢 <s 𝑤 ↔ ∃𝑤 ∈ 𝐴 𝑈 <s 𝑤))
42	41	rspcv 3602	. . . . 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 1223	. . . . . . . . . 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 3978	. . . . . . . . 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 480	. . . . . . . 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 480	. . . . . . . . 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 768	. . . . . . . . 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 3978	. . . . . . . 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 27591	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
51	50	3ad2ant2 1131	. . . . . . . . . . 11 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → 𝑆 ∈ No )
52	51	adantr 480	. . . . . . . . . 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 480	. . . . . . . . 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 27538	. . . . . . . . 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 1226	. . . . . . . . . 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 480	. . . . . . . . 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 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)) → ¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
59		simpll2 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)) → (𝐴 ⊆ No ∧ 𝐴 ∈ V))
60		simpll3 1211	. . . . . . . . . 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 770	. . . . . . . . . . . 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 682	. . . . . . . . . . . 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 511	. . . . . . . . . 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 1371	. . . . . . . . 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 2769	. . . . . . . 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 766	. . . . . . . 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 27583	. . . . . . . 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 1389	. . . . . . 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 456	. . . . . 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 550	. . . . 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 3162	. . . 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 813	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → ¬ (𝑈‘dom 𝑆) = ∅)
75	74	neqned 2941	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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2703   ≠ wne 2934  ∀wral 3055  ∃wrex 3064  Vcvv 3468   ∪ cun 3941   ⊆ wss 3943  ∅c0 4317  ifcif 4523  {csn 4623  ⟨cop 4629   class class class wbr 5141   ↦ cmpt 5224   Or wor 5580  dom cdm 5669   ↾ cres 5671  Oncon0 6358  suc csuc 6360  ℩cio 6487  ‘cfv 6537  ℩crio 7360  2_oc2o 8461   No csur 27528   <s cslt 27529

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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-1o 8467  df-2o 8468  df-no 27531  df-slt 27532  df-bday 27533

This theorem is referenced by:  nosupbnd1lem5  27600  nosupbnd1lem6  27601