Theorem noresle 27623

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 4180	. . . 4 ⊢ ((dom 𝑈 ⊆ 𝐴 ∧ dom 𝑆 ⊆ 𝐴) ↔ (dom 𝑈 ∪ dom 𝑆) ⊆ 𝐴)
2		ssralv 4046	. . . 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 216	. . 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 1115	. 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 5145	. . . . . . . 8 ⊢ (𝑈 = 𝑆 → (𝑈 <s 𝑈 ↔ 𝑆 <s 𝑈))
6	5	notbid 318	. . . . . . 7 ⊢ (𝑈 = 𝑆 → (¬ 𝑈 <s 𝑈 ↔ ¬ 𝑆 <s 𝑈))
7	6	biimpd 228	. . . . . 6 ⊢ (𝑈 = 𝑆 → (¬ 𝑈 <s 𝑈 → ¬ 𝑆 <s 𝑈))
8		sltso 27602	. . . . . . . 8 ⊢ <s Or No
9		sonr 5607	. . . . . . . 8 ⊢ (( <s Or No ∧ 𝑈 ∈ No ) → ¬ 𝑈 <s 𝑈)
10	8, 9	mpan 689	. . . . . . 7 ⊢ (𝑈 ∈ No → ¬ 𝑈 <s 𝑈)
11	10	adantr 480	. . . . . 6 ⊢ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) → ¬ 𝑈 <s 𝑈)
12	7, 11	impel 505	. . . . 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 412	. . 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 482	. . . . . . . . . . 11 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑈 ≠ 𝑆)
19		nosepne 27606	. . . . . . . . . . 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 1369	. . . . . . . . . 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 27591	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ No ∧ 𝑆 ∈ No ∧ 𝑈 ≠ 𝑆) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ On)
22	16, 17, 18, 21	syl3anc 1369	. . . . . . . . . . . 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 6444	. . . . . . . . . . . 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 6911	. . . . . . . . . 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 6911	. . . . . . . . . 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 3014	. . . . . . . . 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 2941	. . . . . . . 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 6890	. . . . . . . 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 140	. . . . . . 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 27610	. . . . . . . . 9 ⊢ ((𝑈 ∈ No ∧ 𝑆 ∈ No ∧ 𝑈 ≠ 𝑆) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ (dom 𝑈 ∪ dom 𝑆))
32	16, 17, 18, 31	syl3anc 1369	. . . . . . . 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 6429	. . . . . . . . . . . 12 ⊢ (𝑔 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} → suc 𝑔 = suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)})
35	34	reseq2d 5979	. . . . . . . . . . 11 ⊢ (𝑔 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} → (𝑆 ↾ suc 𝑔) = (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}))
36	34	reseq2d 5979	. . . . . . . . . . 11 ⊢ (𝑔 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} → (𝑈 ↾ suc 𝑔) = (𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}))
37	35, 36	breq12d 5155	. . . . . . . . . 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 318	. . . . . . . . 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 3604	. . . . . . . 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		onsuc 7808	. . . . . . . . . 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 27586	. . . . . . . . 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 583	. . . . . . . 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 27586	. . . . . . . . 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 583	. . . . . . . 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 5609	. . . . . . . . 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 583	. . . . . . 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 27588	. . . . . . 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 1369	. . . . . 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 5615	. . . . . 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 412	. . 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 3021	. 2 ⊢ (((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)) → ¬ 𝑆 <s 𝑈)
59	4, 58	sylan2 592	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 395   ∨ w3o 1084   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ≠ wne 2936  ∀wral 3057  {crab 3428   ∪ cun 3943   ⊆ wss 3945  ∩ cint 4944   class class class wbr 5142   Or wor 5583  dom cdm 5672   ↾ cres 5674  Oncon0 6363  suc csuc 6365  ‘cfv 6542   No csur 27566   <s cslt 27567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-1o 8480  df-2o 8481  df-no 27569  df-slt 27570

This theorem is referenced by:  nosupbnd1lem1  27634  nosupbnd2  27642  noinfbnd1lem1  27649  noinfbnd2  27657