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Theorem noresle 27650
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 𝑈𝐴 ∧ dom 𝑆𝐴 ∧ ∀𝑔𝐴 ¬ (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.
StepHypRef Expression
1 unss 4186 . . . 4 ((dom 𝑈𝐴 ∧ dom 𝑆𝐴) ↔ (dom 𝑈 ∪ dom 𝑆) ⊆ 𝐴)
2 ssralv 4050 . . . 4 ((dom 𝑈 ∪ dom 𝑆) ⊆ 𝐴 → (∀𝑔𝐴 ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔) → ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)))
31, 2sylbi 216 . . 3 ((dom 𝑈𝐴 ∧ dom 𝑆𝐴) → (∀𝑔𝐴 ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔) → ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)))
433impia 1114 . 2 ((dom 𝑈𝐴 ∧ dom 𝑆𝐴 ∧ ∀𝑔𝐴 ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)) → ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))
5 breq1 5155 . . . . . . . 8 (𝑈 = 𝑆 → (𝑈 <s 𝑈𝑆 <s 𝑈))
65notbid 317 . . . . . . 7 (𝑈 = 𝑆 → (¬ 𝑈 <s 𝑈 ↔ ¬ 𝑆 <s 𝑈))
76biimpd 228 . . . . . 6 (𝑈 = 𝑆 → (¬ 𝑈 <s 𝑈 → ¬ 𝑆 <s 𝑈))
8 sltso 27629 . . . . . . . 8 <s Or No
9 sonr 5617 . . . . . . . 8 (( <s Or No 𝑈 No ) → ¬ 𝑈 <s 𝑈)
108, 9mpan 688 . . . . . . 7 (𝑈 No → ¬ 𝑈 <s 𝑈)
1110adantr 479 . . . . . 6 ((𝑈 No 𝑆 No ) → ¬ 𝑈 <s 𝑈)
127, 11impel 504 . . . . 5 ((𝑈 = 𝑆 ∧ (𝑈 No 𝑆 No )) → ¬ 𝑆 <s 𝑈)
1312adantrr 715 . . . 4 ((𝑈 = 𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ¬ 𝑆 <s 𝑈)
1413ex 411 . . 3 (𝑈 = 𝑆 → (((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)) → ¬ 𝑆 <s 𝑈))
15 simprl 769 . . . . 5 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → (𝑈 No 𝑆 No ))
16 simprll 777 . . . . . . . . . . 11 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑈 No )
17 simprlr 778 . . . . . . . . . . 11 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑆 No )
18 simpl 481 . . . . . . . . . . 11 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑈𝑆)
19 nosepne 27633 . . . . . . . . . . 11 ((𝑈 No 𝑆 No 𝑈𝑆) → (𝑈 {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ≠ (𝑆 {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}))
2016, 17, 18, 19syl3anc 1368 . . . . . . . . . 10 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → (𝑈 {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ≠ (𝑆 {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}))
21 nosepon 27618 . . . . . . . . . . . . 13 ((𝑈 No 𝑆 No 𝑈𝑆) → {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ On)
2216, 17, 18, 21syl3anc 1368 . . . . . . . . . . . 12 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ On)
23 sucidg 6455 . . . . . . . . . . . 12 ( {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ On → {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)})
2422, 23syl 17 . . . . . . . . . . 11 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)})
2524fvresd 6922 . . . . . . . . . 10 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ((𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)})‘ {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) = (𝑈 {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}))
2624fvresd 6922 . . . . . . . . . 10 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ((𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)})‘ {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) = (𝑆 {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}))
2720, 25, 263netr4d 3015 . . . . . . . . 9 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ((𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)})‘ {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ≠ ((𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)})‘ {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}))
2827neneqd 2942 . . . . . . . 8 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ¬ ((𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)})‘ {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) = ((𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)})‘ {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}))
29 fveq1 6901 . . . . . . . 8 ((𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) = (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) → ((𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)})‘ {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) = ((𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)})‘ {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}))
3028, 29nsyl 140 . . . . . . 7 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ¬ (𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) = (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}))
31 nosepdm 27637 . . . . . . . . 9 ((𝑈 No 𝑆 No 𝑈𝑆) → {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ (dom 𝑈 ∪ dom 𝑆))
3216, 17, 18, 31syl3anc 1368 . . . . . . . 8 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ (dom 𝑈 ∪ dom 𝑆))
33 simprr 771 . . . . . . . 8 ((𝑈𝑆 ∧ ((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 6440 . . . . . . . . . . . 12 (𝑔 = {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} → suc 𝑔 = suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)})
3534reseq2d 5989 . . . . . . . . . . 11 (𝑔 = {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} → (𝑆 ↾ suc 𝑔) = (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}))
3634reseq2d 5989 . . . . . . . . . . 11 (𝑔 = {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} → (𝑈 ↾ suc 𝑔) = (𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}))
3735, 36breq12d 5165 . . . . . . . . . 10 (𝑔 = {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} → ((𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔) ↔ (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) <s (𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)})))
3837notbid 317 . . . . . . . . 9 (𝑔 = {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} → (¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔) ↔ ¬ (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) <s (𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)})))
3938rspcv 3607 . . . . . . . 8 ( {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ (dom 𝑈 ∪ dom 𝑆) → (∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔) → ¬ (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) <s (𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)})))
4032, 33, 39sylc 65 . . . . . . 7 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ¬ (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) <s (𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}))
41 onsuc 7820 . . . . . . . . . 10 ( {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ On → suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ On)
4222, 41syl 17 . . . . . . . . 9 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ On)
43 noreson 27613 . . . . . . . . 9 ((𝑈 No ∧ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ On) → (𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ∈ No )
4416, 42, 43syl2anc 582 . . . . . . . 8 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → (𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ∈ No )
45 noreson 27613 . . . . . . . . 9 ((𝑆 No ∧ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ On) → (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ∈ No )
4617, 42, 45syl2anc 582 . . . . . . . 8 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ∈ No )
47 solin 5619 . . . . . . . . 9 (( <s Or No ∧ ((𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ∈ No ∧ (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ∈ No )) → ((𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) <s (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ∨ (𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) = (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ∨ (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) <s (𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)})))
488, 47mpan 688 . . . . . . . 8 (((𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ∈ No ∧ (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ∈ No ) → ((𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) <s (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ∨ (𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) = (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ∨ (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) <s (𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)})))
4944, 46, 48syl2anc 582 . . . . . . 7 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ((𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) <s (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ∨ (𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) = (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ∨ (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) <s (𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)})))
5030, 40, 49ecase23d 1469 . . . . . 6 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → (𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) <s (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}))
51 sltres 27615 . . . . . . 7 ((𝑈 No 𝑆 No ∧ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ On) → ((𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) <s (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) → 𝑈 <s 𝑆))
5216, 17, 42, 51syl3anc 1368 . . . . . 6 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ((𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) <s (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) → 𝑈 <s 𝑆))
5350, 52mpd 15 . . . . 5 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑈 <s 𝑆)
54 soasym 5625 . . . . . 6 (( <s Or No ∧ (𝑈 No 𝑆 No )) → (𝑈 <s 𝑆 → ¬ 𝑆 <s 𝑈))
558, 54mpan 688 . . . . 5 ((𝑈 No 𝑆 No ) → (𝑈 <s 𝑆 → ¬ 𝑆 <s 𝑈))
5615, 53, 55sylc 65 . . . 4 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ¬ 𝑆 <s 𝑈)
5756ex 411 . . 3 (𝑈𝑆 → (((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)) → ¬ 𝑆 <s 𝑈))
5814, 57pm2.61ine 3022 . 2 (((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)) → ¬ 𝑆 <s 𝑈)
594, 58sylan2 591 1 (((𝑈 No 𝑆 No ) ∧ (dom 𝑈𝐴 ∧ dom 𝑆𝐴 ∧ ∀𝑔𝐴 ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ¬ 𝑆 <s 𝑈)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  w3o 1083  w3a 1084   = wceq 1533  wcel 2098  wne 2937  wral 3058  {crab 3430  cun 3947  wss 3949   cint 4953   class class class wbr 5152   Or wor 5593  dom cdm 5682  cres 5684  Oncon0 6374  suc csuc 6376  cfv 6553   No csur 27593   <s cslt 27594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-1o 8493  df-2o 8494  df-no 27596  df-slt 27597
This theorem is referenced by:  nosupbnd1lem1  27661  nosupbnd2  27669  noinfbnd1lem1  27676  noinfbnd2  27684
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