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Theorem noresle 27757
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 4200 . . . 4 ((dom 𝑈𝐴 ∧ dom 𝑆𝐴) ↔ (dom 𝑈 ∪ dom 𝑆) ⊆ 𝐴)
2 ssralv 4064 . . . 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 217 . . 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 1116 . 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 5151 . . . . . . . 8 (𝑈 = 𝑆 → (𝑈 <s 𝑈𝑆 <s 𝑈))
65notbid 318 . . . . . . 7 (𝑈 = 𝑆 → (¬ 𝑈 <s 𝑈 ↔ ¬ 𝑆 <s 𝑈))
76biimpd 229 . . . . . 6 (𝑈 = 𝑆 → (¬ 𝑈 <s 𝑈 → ¬ 𝑆 <s 𝑈))
8 sltso 27736 . . . . . . . 8 <s Or No
9 sonr 5621 . . . . . . . 8 (( <s Or No 𝑈 No ) → ¬ 𝑈 <s 𝑈)
108, 9mpan 690 . . . . . . 7 (𝑈 No → ¬ 𝑈 <s 𝑈)
1110adantr 480 . . . . . 6 ((𝑈 No 𝑆 No ) → ¬ 𝑈 <s 𝑈)
127, 11impel 505 . . . . 5 ((𝑈 = 𝑆 ∧ (𝑈 No 𝑆 No )) → ¬ 𝑆 <s 𝑈)
1312adantrr 717 . . . 4 ((𝑈 = 𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ¬ 𝑆 <s 𝑈)
1413ex 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 771 . . . . 5 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → (𝑈 No 𝑆 No ))
16 simprll 779 . . . . . . . . . . 11 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑈 No )
17 simprlr 780 . . . . . . . . . . 11 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑆 No )
18 simpl 482 . . . . . . . . . . 11 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑈𝑆)
19 nosepne 27740 . . . . . . . . . . 11 ((𝑈 No 𝑆 No 𝑈𝑆) → (𝑈 {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ≠ (𝑆 {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}))
2016, 17, 18, 19syl3anc 1370 . . . . . . . . . 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 27725 . . . . . . . . . . . . 13 ((𝑈 No 𝑆 No 𝑈𝑆) → {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ On)
2216, 17, 18, 21syl3anc 1370 . . . . . . . . . . . 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 6467 . . . . . . . . . . . 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 6927 . . . . . . . . . 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 6927 . . . . . . . . . 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 3016 . . . . . . . . 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 2943 . . . . . . . 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 6906 . . . . . . . 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 27744 . . . . . . . . 9 ((𝑈 No 𝑆 No 𝑈𝑆) → {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ (dom 𝑈 ∪ dom 𝑆))
3216, 17, 18, 31syl3anc 1370 . . . . . . . 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 773 . . . . . . . 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 6452 . . . . . . . . . . . 12 (𝑔 = {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} → suc 𝑔 = suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)})
3534reseq2d 6000 . . . . . . . . . . 11 (𝑔 = {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} → (𝑆 ↾ suc 𝑔) = (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}))
3634reseq2d 6000 . . . . . . . . . . 11 (𝑔 = {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} → (𝑈 ↾ suc 𝑔) = (𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}))
3735, 36breq12d 5161 . . . . . . . . . 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 318 . . . . . . . . 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 3618 . . . . . . . 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 7831 . . . . . . . . . 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 27720 . . . . . . . . 9 ((𝑈 No ∧ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ On) → (𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ∈ No )
4416, 42, 43syl2anc 584 . . . . . . . 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 27720 . . . . . . . . 9 ((𝑆 No ∧ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ On) → (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ∈ No )
4617, 42, 45syl2anc 584 . . . . . . . 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 5623 . . . . . . . . 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 690 . . . . . . . 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 584 . . . . . . 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 1472 . . . . . 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 27722 . . . . . . 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 1370 . . . . . 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 5629 . . . . . 6 (( <s Or No ∧ (𝑈 No 𝑆 No )) → (𝑈 <s 𝑆 → ¬ 𝑆 <s 𝑈))
558, 54mpan 690 . . . . 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 412 . . 3 (𝑈𝑆 → (((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)) → ¬ 𝑆 <s 𝑈))
5814, 57pm2.61ine 3023 . 2 (((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)) → ¬ 𝑆 <s 𝑈)
594, 58sylan2 593 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 395  w3o 1085  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  {crab 3433  cun 3961  wss 3963   cint 4951   class class class wbr 5148   Or wor 5596  dom cdm 5689  cres 5691  Oncon0 6386  suc csuc 6388  cfv 6563   No csur 27699   <s cslt 27700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703
This theorem is referenced by:  nosupbnd1lem1  27768  nosupbnd2  27776  noinfbnd1lem1  27783  noinfbnd2  27791
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