Theorem noinfprefixmo 27933

Description: In any class of surreals, there is at most one value of the prefix property. (Contributed by Scott Fenton, 8-Aug-2024.)

Assertion

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

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

Proof of Theorem noinfprefixmo

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

Step	Hyp	Ref	Expression
1		reeanv 3236	. . . 4 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑝 ∈ 𝐴 ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)) ↔ (∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ ∃𝑝 ∈ 𝐴 (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))
2		breq2 5111	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑝 → (𝑢 <s 𝑣 ↔ 𝑢 <s 𝑝))
3	2	notbid 321	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑝 → (¬ 𝑢 <s 𝑣 ↔ ¬ 𝑢 <s 𝑝))
4		reseq1 5970	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑝 → (𝑣 ↾ suc 𝐺) = (𝑝 ↾ suc 𝐺))
5	4	eqeq2d 2773	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑝 → ((𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺) ↔ (𝑢 ↾ suc 𝐺) = (𝑝 ↾ suc 𝐺)))
6	3, 5	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑝 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ↔ (¬ 𝑢 <s 𝑝 → (𝑢 ↾ suc 𝐺) = (𝑝 ↾ suc 𝐺))))
7		simprl2 1238	. . . . . . . . . . . 12 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦))) → ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
8	7	adantl 487	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
9		simprlr 792	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → 𝑝 ∈ 𝐴)
10	6, 8, 9	rspcdva 3580	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → (¬ 𝑢 <s 𝑝 → (𝑢 ↾ suc 𝐺) = (𝑝 ↾ suc 𝐺)))
11		breq2 5111	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑢 → (𝑝 <s 𝑣 ↔ 𝑝 <s 𝑢))
12	11	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑢 → (¬ 𝑝 <s 𝑣 ↔ ¬ 𝑝 <s 𝑢))
13		reseq1 5970	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑢 → (𝑣 ↾ suc 𝐺) = (𝑢 ↾ suc 𝐺))
14	13	eqeq2d 2773	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑢 → ((𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺) ↔ (𝑝 ↾ suc 𝐺) = (𝑢 ↾ suc 𝐺)))
15	12, 14	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑢 → ((¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ↔ (¬ 𝑝 <s 𝑢 → (𝑝 ↾ suc 𝐺) = (𝑢 ↾ suc 𝐺))))
16		simprr2 1241	. . . . . . . . . . . . 13 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦))) → ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
17	16	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
18		simprll 791	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → 𝑢 ∈ 𝐴)
19	15, 17, 18	rspcdva 3580	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → (¬ 𝑝 <s 𝑢 → (𝑝 ↾ suc 𝐺) = (𝑢 ↾ suc 𝐺)))
20		eqcom 2769	. . . . . . . . . . 11 ⊢ ((𝑝 ↾ suc 𝐺) = (𝑢 ↾ suc 𝐺) ↔ (𝑢 ↾ suc 𝐺) = (𝑝 ↾ suc 𝐺))
21	19, 20	imbitrdi 254	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → (¬ 𝑝 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑝 ↾ suc 𝐺)))
22		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → 𝐴 ⊆ No )
23	22, 18	sseldd 3935	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → 𝑢 ∈ No )
24	22, 9	sseldd 3935	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → 𝑝 ∈ No )
25		ltsso 27908	. . . . . . . . . . . . 13 ⊢ <s Or No
26		soasym 5600	. . . . . . . . . . . . 13 ⊢ (( <s Or No ∧ (𝑢 ∈ No ∧ 𝑝 ∈ No )) → (𝑢 <s 𝑝 → ¬ 𝑝 <s 𝑢))
27	25, 26	mpan 703	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ No ∧ 𝑝 ∈ No ) → (𝑢 <s 𝑝 → ¬ 𝑝 <s 𝑢))
28	23, 24, 27	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → (𝑢 <s 𝑝 → ¬ 𝑝 <s 𝑢))
29		imor 867	. . . . . . . . . . 11 ⊢ ((𝑢 <s 𝑝 → ¬ 𝑝 <s 𝑢) ↔ (¬ 𝑢 <s 𝑝 ∨ ¬ 𝑝 <s 𝑢))
30	28, 29	sylib 221	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → (¬ 𝑢 <s 𝑝 ∨ ¬ 𝑝 <s 𝑢))
31	10, 21, 30	mpjaod 874	. . . . . . . . 9 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → (𝑢 ↾ suc 𝐺) = (𝑝 ↾ suc 𝐺))
32	31	fveq1d 6884	. . . . . . . 8 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → ((𝑢 ↾ suc 𝐺)‘𝐺) = ((𝑝 ↾ suc 𝐺)‘𝐺))
33		simprl1 1237	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦))) → 𝐺 ∈ dom 𝑢)
34	33	adantl 487	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → 𝐺 ∈ dom 𝑢)
35		sucidg 6445	. . . . . . . . . 10 ⊢ (𝐺 ∈ dom 𝑢 → 𝐺 ∈ suc 𝐺)
36	34, 35	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → 𝐺 ∈ suc 𝐺)
37	36	fvresd 6902	. . . . . . . 8 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → ((𝑢 ↾ suc 𝐺)‘𝐺) = (𝑢‘𝐺))
38	36	fvresd 6902	. . . . . . . 8 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → ((𝑝 ↾ suc 𝐺)‘𝐺) = (𝑝‘𝐺))
39	32, 37, 38	3eqtr3d 2805	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → (𝑢‘𝐺) = (𝑝‘𝐺))
40		simprl3 1239	. . . . . . . 8 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦))) → (𝑢‘𝐺) = 𝑥)
41	40	adantl 487	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → (𝑢‘𝐺) = 𝑥)
42		simprr3 1242	. . . . . . . 8 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦))) → (𝑝‘𝐺) = 𝑦)
43	42	adantl 487	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → (𝑝‘𝐺) = 𝑦)
44	39, 41, 43	3eqtr3d 2805	. . . . . 6 ⊢ ((𝐴 ⊆ No ∧ ((𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))) → 𝑥 = 𝑦)
45	44	expr 462	. . . . 5 ⊢ ((𝐴 ⊆ No ∧ (𝑢 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴)) → (((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)) → 𝑥 = 𝑦))
46	45	rexlimdvva 3221	. . . 4 ⊢ (𝐴 ⊆ No → (∃𝑢 ∈ 𝐴 ∃𝑝 ∈ 𝐴 ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)) → 𝑥 = 𝑦))
47	1, 46	biimtrrid 246	. . 3 ⊢ (𝐴 ⊆ No → ((∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ ∃𝑝 ∈ 𝐴 (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)) → 𝑥 = 𝑦))
48	47	alrimivv 1961	. 2 ⊢ (𝐴 ⊆ No → ∀𝑥∀𝑦((∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ ∃𝑝 ∈ 𝐴 (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)) → 𝑥 = 𝑦))
49		eqeq2 2774	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑢‘𝐺) = 𝑥 ↔ (𝑢‘𝐺) = 𝑦))
50	49	3anbi3d 1470	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ↔ (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑦)))
51	50	rexbidv 3188	. . . 4 ⊢ (𝑥 = 𝑦 → (∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ↔ ∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑦)))
52		dmeq 5891	. . . . . . 7 ⊢ (𝑢 = 𝑝 → dom 𝑢 = dom 𝑝)
53	52	eleq2d 2848	. . . . . 6 ⊢ (𝑢 = 𝑝 → (𝐺 ∈ dom 𝑢 ↔ 𝐺 ∈ dom 𝑝))
54		breq1 5110	. . . . . . . . 9 ⊢ (𝑢 = 𝑝 → (𝑢 <s 𝑣 ↔ 𝑝 <s 𝑣))
55	54	notbid 321	. . . . . . . 8 ⊢ (𝑢 = 𝑝 → (¬ 𝑢 <s 𝑣 ↔ ¬ 𝑝 <s 𝑣))
56		reseq1 5970	. . . . . . . . 9 ⊢ (𝑢 = 𝑝 → (𝑢 ↾ suc 𝐺) = (𝑝 ↾ suc 𝐺))
57	56	eqeq1d 2764	. . . . . . . 8 ⊢ (𝑢 = 𝑝 → ((𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺) ↔ (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))
58	55, 57	imbi12d 347	. . . . . . 7 ⊢ (𝑢 = 𝑝 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ↔ (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))))
59	58	ralbidv 3187	. . . . . 6 ⊢ (𝑢 = 𝑝 → (∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ↔ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺))))
60		fveq1 6881	. . . . . . 7 ⊢ (𝑢 = 𝑝 → (𝑢‘𝐺) = (𝑝‘𝐺))
61	60	eqeq1d 2764	. . . . . 6 ⊢ (𝑢 = 𝑝 → ((𝑢‘𝐺) = 𝑦 ↔ (𝑝‘𝐺) = 𝑦))
62	53, 59, 61	3anbi123d 1464	. . . . 5 ⊢ (𝑢 = 𝑝 → ((𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑦) ↔ (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))
63	62	cbvrexvw 3243	. . . 4 ⊢ (∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑦) ↔ ∃𝑝 ∈ 𝐴 (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦))
64	51, 63	bitrdi 290	. . 3 ⊢ (𝑥 = 𝑦 → (∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ↔ ∃𝑝 ∈ 𝐴 (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)))
65	64	mo4 2593	. 2 ⊢ (∃*𝑥∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ↔ ∀𝑥∀𝑦((∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥) ∧ ∃𝑝 ∈ 𝐴 (𝐺 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑝 <s 𝑣 → (𝑝 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑝‘𝐺) = 𝑦)) → 𝑥 = 𝑦))
66	48, 65	sylibr 237	1 ⊢ (𝐴 ⊆ No → ∃*𝑥∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∃*wmo 2564  ∀wral 3078  ∃wrex 3088   ⊆ wss 3902   class class class wbr 5107   Or wor 5566  dom cdm 5659   ↾ cres 5661  suc csuc 6363  ‘cfv 6537   No csur 27872   <s clts 27873

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-1o 8458  df-2o 8459  df-no 27875  df-lts 27876

This theorem is used by:  noinfno  27950  noinffv  27953