Theorem noresle 33900

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 4118	. . . 4 ⊢ ((dom 𝑈 ⊆ 𝐴 ∧ dom 𝑆 ⊆ 𝐴) ↔ (dom 𝑈 ∪ dom 𝑆) ⊆ 𝐴)
2		ssralv 3987	. . . 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 1116	. 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 5077	. . . . . . . 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 33879	. . . . . . . 8 ⊢ <s Or No
9		sonr 5526	. . . . . . . 8 ⊢ (( <s Or No ∧ 𝑈 ∈ No ) → ¬ 𝑈 <s 𝑈)
10	8, 9	mpan 687	. . . . . . 7 ⊢ (𝑈 ∈ No → ¬ 𝑈 <s 𝑈)
11	10	adantr 481	. . . . . 6 ⊢ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) → ¬ 𝑈 <s 𝑈)
12	7, 11	impel 506	. . . . 5 ⊢ ((𝑈 = 𝑆 ∧ (𝑈 ∈ No ∧ 𝑆 ∈ No )) → ¬ 𝑆 <s 𝑈)
13	12	adantrr 714	. . . 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 413	. . 3 ⊢ (𝑈 = 𝑆 → (((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)) → ¬ 𝑆 <s 𝑈))
15		simprl 768	. . . . 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 776	. . . . . . . . . . 11 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑈 ∈ No )
17		simprlr 777	. . . . . . . . . . 11 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑆 ∈ No )
18		simpl 483	. . . . . . . . . . 11 ⊢ ((𝑈 ≠ 𝑆 ∧ ((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑈 ≠ 𝑆)
19		nosepne 33883	. . . . . . . . . . 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 1370	. . . . . . . . . 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 33868	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ No ∧ 𝑆 ∈ No ∧ 𝑈 ≠ 𝑆) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ On)
22	16, 17, 18, 21	syl3anc 1370	. . . . . . . . . . . 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 6344	. . . . . . . . . . . 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 6794	. . . . . . . . . 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 6794	. . . . . . . . . 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 3021	. . . . . . . . 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 2948	. . . . . . . 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 6773	. . . . . . . 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 33887	. . . . . . . . 9 ⊢ ((𝑈 ∈ No ∧ 𝑆 ∈ No ∧ 𝑈 ≠ 𝑆) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} ∈ (dom 𝑈 ∪ dom 𝑆))
32	16, 17, 18, 31	syl3anc 1370	. . . . . . . 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 770	. . . . . . . 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 6331	. . . . . . . . . . . 12 ⊢ (𝑔 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} → suc 𝑔 = suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)})
35	34	reseq2d 5891	. . . . . . . . . . 11 ⊢ (𝑔 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} → (𝑆 ↾ suc 𝑔) = (𝑆 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}))
36	34	reseq2d 5891	. . . . . . . . . . 11 ⊢ (𝑔 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)} → (𝑈 ↾ suc 𝑔) = (𝑈 ↾ suc ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑆‘𝑥)}))
37	35, 36	breq12d 5087	. . . . . . . . . 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 3557	. . . . . . . 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 7659	. . . . . . . . . 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 33863	. . . . . . . . 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 33863	. . . . . . . . 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 5528	. . . . . . . . 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 687	. . . . . . . 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 1472	. . . . . 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 33865	. . . . . . 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 1370	. . . . . 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 5534	. . . . . 6 ⊢ (( <s Or No ∧ (𝑈 ∈ No ∧ 𝑆 ∈ No )) → (𝑈 <s 𝑆 → ¬ 𝑆 <s 𝑈))
55	8, 54	mpan 687	. . . . 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 413	. . 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 3028	. 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 396   ∨ w3o 1085   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  {crab 3068   ∪ cun 3885   ⊆ wss 3887  ∩ cint 4879   class class class wbr 5074   Or wor 5502  dom cdm 5589   ↾ cres 5591  Oncon0 6266  suc csuc 6268  ‘cfv 6433   No csur 33843   <s cslt 33844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-1o 8297  df-2o 8298  df-no 33846  df-slt 33847

This theorem is referenced by:  nosupbnd1lem1  33911  nosupbnd2  33919  noinfbnd1lem1  33926  noinfbnd2  33934