Theorem nosupprefixmo 27666

Description: In any class of surreals, there is at most one value of the prefix property. (Contributed by Scott Fenton, 26-Nov-2021.)

Assertion

Ref	Expression
nosupprefixmo	⊢ (𝐴 ⊆ No → ∃*𝑥∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥))

Distinct variable groups:   𝑢,𝐴,𝑣,𝑥   𝑢,𝐺,𝑣,𝑥

Proof of Theorem nosupprefixmo

Dummy variables 𝑦 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reeanv 3206	. . . 4 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑝 ∈ 𝐴 ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)) ↔ (∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ ∃𝑝 ∈ 𝐴 (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))
2		breq1 5099	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑢 → (𝑣 <s 𝑝 ↔ 𝑢 <s 𝑝))
3	2	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑢 → (¬ 𝑣 <s 𝑝 ↔ ¬ 𝑢 <s 𝑝))
4		reseq1 5930	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑢 → (𝑣 ↾ suc 𝐺) = (𝑢 ↾ suc 𝐺))
5	4	eqeq2d 2745	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑢 → ((𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺) ↔ (𝑝 ↾ suc 𝐺) = (𝑢 ↾ suc 𝐺)))
6	3, 5	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑢 → ((¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ↔ (¬ 𝑢 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑢 ↾ suc 𝐺))))
7		simprr2 1223	. . . . . . . . . . . . 13 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦))) → ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
8	7	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
9		simprll 778	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → 𝑢 ∈ 𝐴)
10	6, 8, 9	rspcdva 3575	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → (¬ 𝑢 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑢 ↾ suc 𝐺)))
11		eqcom 2741	. . . . . . . . . . 11 ⊢ ((𝑝 ↾ suc 𝐺) = (𝑢 ↾ suc 𝐺) ↔ (𝑢 ↾ suc 𝐺) = (𝑝 ↾ suc 𝐺))
12	10, 11	imbitrdi 251	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → (¬ 𝑢 <s 𝑝 → (𝑢 ↾ suc 𝐺) = (𝑝 ↾ suc 𝐺)))
13		breq1 5099	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑝 → (𝑣 <s 𝑢 ↔ 𝑝 <s 𝑢))
14	13	notbid 318	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑝 → (¬ 𝑣 <s 𝑢 ↔ ¬ 𝑝 <s 𝑢))
15		reseq1 5930	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑝 → (𝑣 ↾ suc 𝐺) = (𝑝 ↾ suc 𝐺))
16	15	eqeq2d 2745	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑝 → ((𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺) ↔ (𝑢 ↾ suc 𝐺) = (𝑝 ↾ suc 𝐺)))
17	14, 16	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑝 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ↔ (¬ 𝑝 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑝 ↾ suc 𝐺))))
18		simprl2 1220	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦))) → ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
19	18	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
20		simprlr 779	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → 𝑝 ∈ 𝐴)
21	17, 19, 20	rspcdva 3575	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → (¬ 𝑝 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑝 ↾ suc 𝐺)))
22		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → 𝐴 ⊆ No )
23	22, 9	sseldd 3932	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → 𝑢 ∈ No )
24	22, 20	sseldd 3932	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → 𝑝 ∈ No )
25		sltso 27642	. . . . . . . . . . . . 13 ⊢ <s Or No
26		soasym 5563	. . . . . . . . . . . . 13 ⊢ (( <s Or No ∧ (𝑢 ∈ No ∧ 𝑝 ∈ No )) → (𝑢 <s 𝑝 → ¬ 𝑝 <s 𝑢))
27	25, 26	mpan 690	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ No ∧ 𝑝 ∈ No ) → (𝑢 <s 𝑝 → ¬ 𝑝 <s 𝑢))
28	23, 24, 27	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → (𝑢 <s 𝑝 → ¬ 𝑝 <s 𝑢))
29		pm4.62 856	. . . . . . . . . . 11 ⊢ ((𝑢 <s 𝑝 → ¬ 𝑝 <s 𝑢) ↔ (¬ 𝑢 <s 𝑝 ∨ ¬ 𝑝 <s 𝑢))
30	28, 29	sylib 218	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → (¬ 𝑢 <s 𝑝 ∨ ¬ 𝑝 <s 𝑢))
31	12, 21, 30	mpjaod 860	. . . . . . . . 9 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → (𝑢 ↾ suc 𝐺) = (𝑝 ↾ suc 𝐺))
32	31	fveq1d 6834	. . . . . . . 8 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → ((𝑢 ↾ suc 𝐺)‘𝐺) = ((𝑝 ↾ suc 𝐺)‘𝐺))
33		simprl1 1219	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦))) → 𝐺 ∈ dom 𝑢)
34	33	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → 𝐺 ∈ dom 𝑢)
35		sucidg 6398	. . . . . . . . . 10 ⊢ (𝐺 ∈ dom 𝑢 → 𝐺 ∈ suc 𝐺)
36	34, 35	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → 𝐺 ∈ suc 𝐺)
37	36	fvresd 6852	. . . . . . . 8 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → ((𝑢 ↾ suc 𝐺)‘𝐺) = (𝑢‘𝐺))
38	36	fvresd 6852	. . . . . . . 8 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → ((𝑝 ↾ suc 𝐺)‘𝐺) = (𝑝‘𝐺))
39	32, 37, 38	3eqtr3d 2777	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → (𝑢‘𝐺) = (𝑝‘𝐺))
40		simprl3 1221	. . . . . . . 8 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦))) → (𝑢‘𝐺) = 𝑥)
41	40	adantl 481	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → (𝑢‘𝐺) = 𝑥)
42		simprr3 1224	. . . . . . . 8 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦))) → (𝑝‘𝐺) = 𝑦)
43	42	adantl 481	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → (𝑝‘𝐺) = 𝑦)
44	39, 41, 43	3eqtr3d 2777	. . . . . 6 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → 𝑥 = 𝑦)
45	44	expr 456	. . . . 5 ⊢ ((𝐴 ⊆ No ∧ (𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴)) → (((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)) → 𝑥 = 𝑦))
46	45	rexlimdvva 3191	. . . 4 ⊢ (𝐴 ⊆ No → (∃𝑢 ∈ 𝐴 ∃𝑝 ∈ 𝐴 ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)) → 𝑥 = 𝑦))
47	1, 46	biimtrrid 243	. . 3 ⊢ (𝐴 ⊆ No → ((∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ ∃𝑝 ∈ 𝐴 (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)) → 𝑥 = 𝑦))
48	47	alrimivv 1929	. 2 ⊢ (𝐴 ⊆ No → ∀𝑥∀𝑦((∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ ∃𝑝 ∈ 𝐴 (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)) → 𝑥 = 𝑦))
49		eqeq2 2746	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑢‘𝐺) = 𝑥 ↔ (𝑢‘𝐺) = 𝑦))
50	49	3anbi3d 1444	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ↔ (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑦)))
51	50	rexbidv 3158	. . . 4 ⊢ (𝑥 = 𝑦 → (∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ↔ ∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑦)))
52		dmeq 5850	. . . . . . 7 ⊢ (𝑢 = 𝑝 → dom 𝑢 = dom 𝑝)
53	52	eleq2d 2820	. . . . . 6 ⊢ (𝑢 = 𝑝 → (𝐺 ∈ dom 𝑢 ↔ 𝐺 ∈ dom 𝑝))
54		breq2 5100	. . . . . . . . 9 ⊢ (𝑢 = 𝑝 → (𝑣 <s 𝑢 ↔ 𝑣 <s 𝑝))
55	54	notbid 318	. . . . . . . 8 ⊢ (𝑢 = 𝑝 → (¬ 𝑣 <s 𝑢 ↔ ¬ 𝑣 <s 𝑝))
56		reseq1 5930	. . . . . . . . 9 ⊢ (𝑢 = 𝑝 → (𝑢 ↾ suc 𝐺) = (𝑝 ↾ suc 𝐺))
57	56	eqeq1d 2736	. . . . . . . 8 ⊢ (𝑢 = 𝑝 → ((𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺) ↔ (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
58	55, 57	imbi12d 344	. . . . . . 7 ⊢ (𝑢 = 𝑝 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ↔ (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))))
59	58	ralbidv 3157	. . . . . 6 ⊢ (𝑢 = 𝑝 → (∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ↔ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))))
60		fveq1 6831	. . . . . . 7 ⊢ (𝑢 = 𝑝 → (𝑢‘𝐺) = (𝑝‘𝐺))
61	60	eqeq1d 2736	. . . . . 6 ⊢ (𝑢 = 𝑝 → ((𝑢‘𝐺) = 𝑦 ↔ (𝑝‘𝐺) = 𝑦))
62	53, 59, 61	3anbi123d 1438	. . . . 5 ⊢ (𝑢 = 𝑝 → ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑦) ↔ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))
63	62	cbvrexvw 3213	. . . 4 ⊢ (∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑦) ↔ ∃𝑝 ∈ 𝐴 (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦))
64	51, 63	bitrdi 287	. . 3 ⊢ (𝑥 = 𝑦 → (∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ↔ ∃𝑝 ∈ 𝐴 (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))
65	64	mo4 2564	. 2 ⊢ (∃*𝑥∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ↔ ∀𝑥∀𝑦((∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ ∃𝑝 ∈ 𝐴 (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)) → 𝑥 = 𝑦))
66	48, 65	sylibr 234	1 ⊢ (𝐴 ⊆ No → ∃*𝑥∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086  ∀wal 1539   = wceq 1541   ∈ wcel 2113  ∃*wmo 2535  ∀wral 3049  ∃wrex 3058   ⊆ wss 3899   class class class wbr 5096   Or wor 5529  dom cdm 5622   ↾ cres 5624  suc csuc 6317  ‘cfv 6490   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-sep 5239  ax-nul 5249  ax-pr 5375

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-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-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-fv 6498  df-1o 8395  df-2o 8396  df-no 27608  df-slt 27609

This theorem is referenced by:  nosupno  27669  nosupfv  27672