Theorem noresle 28111

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 4191	. . . 4 ⊢ ((dom 𝑈 ⊆ 𝐴 ∧ dom 𝑆 ⊆ 𝐴) ↔ (dom 𝑈 ∪ dom 𝑆) ⊆ 𝐴)
2		ssralv 3982	. . . 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 1135	. 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 5111	. . . . . . . 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		ltsso 28090	. . . . . . . 8 ⊢ <s Or No
9		sonr 5590	. . . . . . . 8 ⊢ (( <s Or No ∧ 𝑈 ∈ No) → ¬ 𝑈 <s 𝑈)
10	8, 9	mpan 703	. . . . . . 7 ⊢ (𝑈 ∈ No → ¬ 𝑈 <s 𝑈)
11	10	adantr 486	. . . . . 6 ⊢ ((𝑈 ∈ No ∧ 𝑆 ∈ No) → ¬ 𝑈 <s 𝑈)
12	7, 11	impel 515	. . . . 5 ⊢ ((𝑈 = 𝑆 ∧ (𝑈 ∈ No ∧ 𝑆 ∈ No)) → ¬ 𝑆 <s 𝑈)
13	12	adantrr 730	. . . 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 418	. . 3 ⊢ (𝑈 = 𝑆 → (((𝑈 ∈ No ∧ 𝑆 ∈ No) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)) → ¬ 𝑆 <s 𝑈))
15		simprl 783	. . . . 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 791	. . . . . . . . . . 11 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑈 ∈ No)
17		simprlr 792	. . . . . . . . . . 11 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑆 ∈ No)
18		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑈 ≠ 𝑆)
19		nosepne 28094	. . . . . . . . . . 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 1399	. . . . . . . . . 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 28079	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ No ∧ 𝑆 ∈ No ∧ 𝑈 ≠ 𝑆) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ On)
22	16, 17, 18, 21	syl3anc 1399	. . . . . . . . . . . 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 6472	. . . . . . . . . . . 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 18	. . . . . . . . . . 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 6936	. . . . . . . . . 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 6936	. . . . . . . . . 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 3036	. . . . . . . . 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 2964	. . . . . . . 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 6915	. . . . . . . 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 141	. . . . . . 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 28098	. . . . . . . . 9 ⊢ ((𝑈 ∈ No ∧ 𝑆 ∈ No ∧ 𝑈 ≠ 𝑆) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ (dom 𝑈 ∪ dom 𝑆))
32	16, 17, 18, 31	syl3anc 1399	. . . . . . . 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 785	. . . . . . . 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 6457	. . . . . . . . . . . 12 ⊢ (𝑔 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} → suc 𝑔 = suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)})
35	34	reseq2d 5986	. . . . . . . . . . 11 ⊢ (𝑔 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} → (𝑆 ↾ suc 𝑔) = (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}))
36	34	reseq2d 5986	. . . . . . . . . . 11 ⊢ (𝑔 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} → (𝑈 ↾ suc 𝑔) = (𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}))
37	35, 36	breq12d 5121	. . . . . . . . . 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 3576	. . . . . . . 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 66	. . . . . . 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		onsuc 7858	. . . . . . . . . 10 ⊢ (∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ On → suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ On)
42	22, 41	syl 18	. . . . . . . . 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 28074	. . . . . . . . 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 596	. . . . . . . 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 28074	. . . . . . . . 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 596	. . . . . . . 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 5593	. . . . . . . . 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 703	. . . . . . . 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 596	. . . . . . 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 1504	. . . . . 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		ltsres 28076	. . . . . . 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 1399	. . . . . 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 16	. . . . 5 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑈 <s 𝑆)
54		soasym 5599	. . . . . 6 ⊢ (( <s Or No ∧ (𝑈 ∈ No ∧ 𝑆 ∈ No)) → (𝑈 <s 𝑆 → ¬ 𝑆 <s 𝑈))
55	8, 54	mpan 703	. . . . 5 ⊢ ((𝑈 ∈ No ∧ 𝑆 ∈ No) → (𝑈 <s 𝑆 → ¬ 𝑆 <s 𝑈))
56	15, 53, 55	sylc 66	. . . 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 418	. . 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 3042	. 2 ⊢ (((𝑈 ∈ No ∧ 𝑆 ∈ No) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)) → ¬ 𝑆 <s 𝑈)
59	4, 58	sylan2 605	1 ⊢ (((𝑈 ∈ No ∧ 𝑆 ∈ No) ∧ (dom 𝑈 ⊆ 𝐴 ∧ dom 𝑆 ⊆ 𝐴 ∧ ∀𝑔 ∈ 𝐴 ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ¬ 𝑆 <s 𝑈)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  {crab 3416   ⊆ wss 3900   ∪ cun 4152  ∩ cint 4912   class class class wbr 5108   Or wor 5565  dom cdm 5658   ↾ cres 5660  Oncon0 6388  suc csuc 6390  ‘cfv 6566  Nocsur 28054   <s clts 28055

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-1o 8514  df-2o 8515  df-no 28057  df-lts 28058

This theorem is used by:  nosupbnd1lem1  28122  nosupbnd2  28130  noinfbnd1lem1  28137  noinfbnd2  28145