Theorem noresle 33313

Description: Restriction law for surreals. Lemma 2.1.4 of [Lipparini] p. 3. (Contributed by Scott Fenton, 5-Dec-2021.)

Assertion

Ref	Expression
noresle	⊢ (((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ (dom 𝑈 ⊆ 𝐴 ∧ dom 𝑆 ⊆ 𝐴 ∧ ∀𝑔 ∈ 𝐴 ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ¬ 𝑆 <s 𝑈)

Distinct variable groups:   𝑆,𝑔   𝑈,𝑔   𝐴,𝑔

Proof of Theorem noresle

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unss 4111	. . . 4 ⊢ ((dom 𝑈 ⊆ 𝐴 ∧ dom 𝑆 ⊆ 𝐴) ↔ (dom 𝑈 ∪ dom 𝑆) ⊆ 𝐴)
2		ssralv 3981	. . . 4 ⊢ ((dom 𝑈 ∪ dom 𝑆) ⊆ 𝐴 → (∀𝑔 ∈ 𝐴 ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔) → ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)))
3	1, 2	sylbi 220	. . 3 ⊢ ((dom 𝑈 ⊆ 𝐴 ∧ dom 𝑆 ⊆ 𝐴) → (∀𝑔 ∈ 𝐴 ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔) → ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)))
4	3	3impia 1114	. 2 ⊢ ((dom 𝑈 ⊆ 𝐴 ∧ dom 𝑆 ⊆ 𝐴 ∧ ∀𝑔 ∈ 𝐴 ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)) → ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))
5		breq1 5033	. . . . . . . 8 ⊢ (𝑈 = 𝑆 → (𝑈 <s 𝑈 ↔ 𝑆 <s 𝑈))
6	5	notbid 321	. . . . . . 7 ⊢ (𝑈 = 𝑆 → (¬ 𝑈 <s 𝑈 ↔ ¬ 𝑆 <s 𝑈))
7	6	biimpd 232	. . . . . 6 ⊢ (𝑈 = 𝑆 → (¬ 𝑈 <s 𝑈 → ¬ 𝑆 <s 𝑈))
8		sltso 33294	. . . . . . . 8 ⊢ <s Or No
9		sonr 5460	. . . . . . . 8 ⊢ (( <s Or No ∧ 𝑈 ∈ No ) → ¬ 𝑈 <s 𝑈)
10	8, 9	mpan 689	. . . . . . 7 ⊢ (𝑈 ∈ No → ¬ 𝑈 <s 𝑈)
11	10	adantr 484	. . . . . 6 ⊢ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) → ¬ 𝑈 <s 𝑈)
12	7, 11	impel 509	. . . . 5 ⊢ ((𝑈 = 𝑆 ∧ (𝑈 ∈ No ∧ 𝑆 ∈ No )) → ¬ 𝑆 <s 𝑈)
13	12	adantrr 716	. . . 4 ⊢ ((𝑈 = 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ¬ 𝑆 <s 𝑈)
14	13	ex 416	. . 3 ⊢ (𝑈 = 𝑆 → (((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)) → ¬ 𝑆 <s 𝑈))
15		simprl 770	. . . . 5 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → (𝑈 ∈ No ∧ 𝑆 ∈ No ))
16		simprll 778	. . . . . . . . . . 11 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑈 ∈ No )
17		simprlr 779	. . . . . . . . . . 11 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑆 ∈ No )
18		simpl 486	. . . . . . . . . . 11 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑈 ≠ 𝑆)
19		nosepne 33298	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ No ∧ 𝑆 ∈ No ∧ 𝑈 ≠ 𝑆) → (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) ≠ (𝑆‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}))
20	16, 17, 18, 19	syl3anc 1368	. . . . . . . . . 10 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) ≠ (𝑆‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}))
21		nosepon 33285	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ No ∧ 𝑆 ∈ No ∧ 𝑈 ≠ 𝑆) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ On)
22	16, 17, 18, 21	syl3anc 1368	. . . . . . . . . . . 12 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ On)
23		sucidg 6237	. . . . . . . . . . . 12 ⊢ (∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ On → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)})
24	22, 23	syl 17	. . . . . . . . . . 11 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)})
25	24	fvresd 6665	. . . . . . . . . 10 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ((𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)})‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) = (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}))
26	24	fvresd 6665	. . . . . . . . . 10 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ((𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)})‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) = (𝑆‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}))
27	20, 25, 26	3netr4d 3064	. . . . . . . . 9 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ((𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)})‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) ≠ ((𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)})‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}))
28	27	neneqd 2992	. . . . . . . 8 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ¬ ((𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)})‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) = ((𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)})‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}))
29		fveq1 6644	. . . . . . . 8 ⊢ ((𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) = (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) → ((𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)})‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) = ((𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)})‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}))
30	28, 29	nsyl 142	. . . . . . 7 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ¬ (𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) = (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}))
31		nosepdm 33301	. . . . . . . . 9 ⊢ ((𝑈 ∈ No ∧ 𝑆 ∈ No ∧ 𝑈 ≠ 𝑆) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ (dom 𝑈 ∪ dom 𝑆))
32	16, 17, 18, 31	syl3anc 1368	. . . . . . . 8 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ (dom 𝑈 ∪ dom 𝑆))
33		simprr 772	. . . . . . . 8 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))
34		suceq 6224	. . . . . . . . . . . 12 ⊢ (𝑔 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} → suc 𝑔 = suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)})
35	34	reseq2d 5818	. . . . . . . . . . 11 ⊢ (𝑔 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} → (𝑆 ↾ suc 𝑔) = (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}))
36	34	reseq2d 5818	. . . . . . . . . . 11 ⊢ (𝑔 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} → (𝑈 ↾ suc 𝑔) = (𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}))
37	35, 36	breq12d 5043	. . . . . . . . . 10 ⊢ (𝑔 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} → ((𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔) ↔ (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) <s (𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)})))
38	37	notbid 321	. . . . . . . . 9 ⊢ (𝑔 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} → (¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔) ↔ ¬ (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) <s (𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)})))
39	38	rspcv 3566	. . . . . . . 8 ⊢ (∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ (dom 𝑈 ∪ dom 𝑆) → (∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔) → ¬ (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) <s (𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)})))
40	32, 33, 39	sylc 65	. . . . . . 7 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ¬ (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) <s (𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}))
41		suceloni 7508	. . . . . . . . . 10 ⊢ (∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ On → suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ On)
42	22, 41	syl 17	. . . . . . . . 9 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ On)
43		noreson 33280	. . . . . . . . 9 ⊢ ((𝑈 ∈ No ∧ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ On) → (𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) ∈ No )
44	16, 42, 43	syl2anc 587	. . . . . . . 8 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → (𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) ∈ No )
45		noreson 33280	. . . . . . . . 9 ⊢ ((𝑆 ∈ No ∧ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ On) → (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) ∈ No )
46	17, 42, 45	syl2anc 587	. . . . . . . 8 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) ∈ No )
47		solin 5462	. . . . . . . . 9 ⊢ (( <s Or No ∧ ((𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) ∈ No ∧ (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) ∈ No )) → ((𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) <s (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) ∨ (𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) = (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) ∨ (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) <s (𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)})))
48	8, 47	mpan 689	. . . . . . . 8 ⊢ (((𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) ∈ No ∧ (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) ∈ No ) → ((𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) <s (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) ∨ (𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) = (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) ∨ (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) <s (𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)})))
49	44, 46, 48	syl2anc 587	. . . . . . 7 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ((𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) <s (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) ∨ (𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) = (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) ∨ (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) <s (𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)})))
50	30, 40, 49	ecase23d 1470	. . . . . 6 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → (𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) <s (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}))
51		sltres 33282	. . . . . . 7 ⊢ ((𝑈 ∈ No ∧ 𝑆 ∈ No ∧ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ On) → ((𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) <s (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) → 𝑈 <s 𝑆))
52	16, 17, 42, 51	syl3anc 1368	. . . . . 6 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ((𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) <s (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}) → 𝑈 <s 𝑆))
53	50, 52	mpd 15	. . . . 5 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑈 <s 𝑆)
54		soasym 5468	. . . . . 6 ⊢ (( <s Or No ∧ (𝑈 ∈ No ∧ 𝑆 ∈ No )) → (𝑈 <s 𝑆 → ¬ 𝑆 <s 𝑈))
55	8, 54	mpan 689	. . . . 5 ⊢ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) → (𝑈 <s 𝑆 → ¬ 𝑆 <s 𝑈))
56	15, 53, 55	sylc 65	. . . 4 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ¬ 𝑆 <s 𝑈)
57	56	ex 416	. . 3 ⊢ (𝑈 ≠ 𝑆 → (((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)) → ¬ 𝑆 <s 𝑈))
58	14, 57	pm2.61ine 3070	. 2 ⊢ (((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)) → ¬ 𝑆 <s 𝑈)
59	4, 58	sylan2 595	1 ⊢ (((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ (dom 𝑈 ⊆ 𝐴 ∧ dom 𝑆 ⊆ 𝐴 ∧ ∀𝑔 ∈ 𝐴 ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ¬ 𝑆 <s 𝑈)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ w3o 1083   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  {crab 3110   ∪ cun 3879   ⊆ wss 3881  ∩ cint 4838   class class class wbr 5030   Or wor 5437  dom cdm 5519   ↾ cres 5521  Oncon0 6159  suc csuc 6161  ‘cfv 6324   No csur 33260   <s cslt 33261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-1o 8085  df-2o 8086  df-no 33263  df-slt 33264

This theorem is referenced by:  nosupbnd1lem1  33321  nosupbnd2  33329