Theorem noresle 27665

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 4142	. . . 4 ⊢ ((dom 𝑈 ⊆ 𝐴 ∧ dom 𝑆 ⊆ 𝐴) ↔ (dom 𝑈 ∪ dom 𝑆) ⊆ 𝐴)
2		ssralv 4002	. . . 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 217	. . 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 1117	. 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 5101	. . . . . . . 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 229	. . . . . 6 ⊢ (𝑈 = 𝑆 → (¬ 𝑈 <s 𝑈 → ¬ 𝑆 <s 𝑈))
8		ltsso 27644	. . . . . . . 8 ⊢ <s Or No
9		sonr 5556	. . . . . . . 8 ⊢ (( <s Or No ∧ 𝑈 ∈ No ) → ¬ 𝑈 <s 𝑈)
10	8, 9	mpan 690	. . . . . . 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 717	. . . 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 27648	. . . . . . . . . . 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 1373	. . . . . . . . . 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 27633	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ No ∧ 𝑆 ∈ No ∧ 𝑈 ≠ 𝑆) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ On)
22	16, 17, 18, 21	syl3anc 1373	. . . . . . . . . . . 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 6400	. . . . . . . . . . . 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 6854	. . . . . . . . . 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 6854	. . . . . . . . . 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 3009	. . . . . . . . 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 2937	. . . . . . . 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 6833	. . . . . . . 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 27652	. . . . . . . . 9 ⊢ ((𝑈 ∈ No ∧ 𝑆 ∈ No ∧ 𝑈 ≠ 𝑆) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ (dom 𝑈 ∪ dom 𝑆))
32	16, 17, 18, 31	syl3anc 1373	. . . . . . . 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 6385	. . . . . . . . . . . 12 ⊢ (𝑔 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} → suc 𝑔 = suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)})
35	34	reseq2d 5938	. . . . . . . . . . 11 ⊢ (𝑔 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} → (𝑆 ↾ suc 𝑔) = (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}))
36	34	reseq2d 5938	. . . . . . . . . . 11 ⊢ (𝑔 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} → (𝑈 ↾ suc 𝑔) = (𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}))
37	35, 36	breq12d 5111	. . . . . . . . . 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 3572	. . . . . . . 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 7755	. . . . . . . . . 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 27628	. . . . . . . . 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 584	. . . . . . . 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 27628	. . . . . . . . 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 584	. . . . . . . 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 5559	. . . . . . . . 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 690	. . . . . . . 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 584	. . . . . . 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 1475	. . . . . 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 27630	. . . . . . 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 1373	. . . . . 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 5565	. . . . . 6 ⊢ (( <s Or No ∧ (𝑈 ∈ No ∧ 𝑆 ∈ No )) → (𝑈 <s 𝑆 → ¬ 𝑆 <s 𝑈))
55	8, 54	mpan 690	. . . . 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 3015	. 2 ⊢ (((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)) → ¬ 𝑆 <s 𝑈)
59	4, 58	sylan2 593	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 1085   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  {crab 3399   ∪ cun 3899   ⊆ wss 3901  ∩ cint 4902   class class class wbr 5098   Or wor 5531  dom cdm 5624   ↾ cres 5626  Oncon0 6317  suc csuc 6319  ‘cfv 6492   No csur 27607   <s clts 27608

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 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-1o 8397  df-2o 8398  df-no 27610  df-lts 27611

This theorem is referenced by:  nosupbnd1lem1  27676  nosupbnd2  27684  noinfbnd1lem1  27691  noinfbnd2  27699