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Theorem noresle 32222
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 3949 . . . 4 ((dom 𝑈𝐴 ∧ dom 𝑆𝐴) ↔ (dom 𝑈 ∪ dom 𝑆) ⊆ 𝐴)
2 ssralv 3826 . . . 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 208 . . 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 1145 . 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 4812 . . . . . . . 8 (𝑈 = 𝑆 → (𝑈 <s 𝑈𝑆 <s 𝑈))
65notbid 309 . . . . . . 7 (𝑈 = 𝑆 → (¬ 𝑈 <s 𝑈 ↔ ¬ 𝑆 <s 𝑈))
76biimpd 220 . . . . . 6 (𝑈 = 𝑆 → (¬ 𝑈 <s 𝑈 → ¬ 𝑆 <s 𝑈))
8 sltso 32203 . . . . . . . 8 <s Or No
9 sonr 5219 . . . . . . . 8 (( <s Or No 𝑈 No ) → ¬ 𝑈 <s 𝑈)
108, 9mpan 681 . . . . . . 7 (𝑈 No → ¬ 𝑈 <s 𝑈)
1110adantr 472 . . . . . 6 ((𝑈 No 𝑆 No ) → ¬ 𝑈 <s 𝑈)
127, 11impel 501 . . . . 5 ((𝑈 = 𝑆 ∧ (𝑈 No 𝑆 No )) → ¬ 𝑆 <s 𝑈)
1312adantrr 708 . . . 4 ((𝑈 = 𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ¬ 𝑆 <s 𝑈)
1413ex 401 . . 3 (𝑈 = 𝑆 → (((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)) → ¬ 𝑆 <s 𝑈))
15 simprl 787 . . . . 5 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → (𝑈 No 𝑆 No ))
16 simprll 797 . . . . . . . . . . 11 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑈 No )
17 simprlr 798 . . . . . . . . . . 11 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑆 No )
18 simpl 474 . . . . . . . . . . 11 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑈𝑆)
19 nosepne 32207 . . . . . . . . . . 11 ((𝑈 No 𝑆 No 𝑈𝑆) → (𝑈 {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ≠ (𝑆 {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}))
2016, 17, 18, 19syl3anc 1490 . . . . . . . . . 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 32194 . . . . . . . . . . . . 13 ((𝑈 No 𝑆 No 𝑈𝑆) → {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ On)
2216, 17, 18, 21syl3anc 1490 . . . . . . . . . . . 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 5986 . . . . . . . . . . . 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 6395 . . . . . . . . . 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 6395 . . . . . . . . . 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 3014 . . . . . . . . 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 6374 . . . . . . . 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 137 . . . . . . 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 32210 . . . . . . . . 9 ((𝑈 No 𝑆 No 𝑈𝑆) → {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ (dom 𝑈 ∪ dom 𝑆))
3216, 17, 18, 31syl3anc 1490 . . . . . . . 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 789 . . . . . . . 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 5973 . . . . . . . . . . . 12 (𝑔 = {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} → suc 𝑔 = suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)})
3534reseq2d 5565 . . . . . . . . . . 11 (𝑔 = {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} → (𝑆 ↾ suc 𝑔) = (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}))
3634reseq2d 5565 . . . . . . . . . . 11 (𝑔 = {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} → (𝑈 ↾ suc 𝑔) = (𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}))
3735, 36breq12d 4822 . . . . . . . . . 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 309 . . . . . . . . 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 3457 . . . . . . . 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 suceloni 7211 . . . . . . . . . 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 32189 . . . . . . . . 9 ((𝑈 No ∧ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ On) → (𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ∈ No )
4416, 42, 43syl2anc 579 . . . . . . . 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 32189 . . . . . . . . 9 ((𝑆 No ∧ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ On) → (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ∈ No )
4617, 42, 45syl2anc 579 . . . . . . . 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 5221 . . . . . . . . 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 681 . . . . . . . 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 579 . . . . . . 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 1597 . . . . . 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 32191 . . . . . . 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 1490 . . . . . 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 32034 . . . . . 6 (( <s Or No ∧ (𝑈 No 𝑆 No )) → (𝑈 <s 𝑆 → ¬ 𝑆 <s 𝑈))
558, 54mpan 681 . . . . 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 401 . . 3 (𝑈𝑆 → (((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)) → ¬ 𝑆 <s 𝑈))
5814, 57pm2.61ine 3020 . 2 (((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)) → ¬ 𝑆 <s 𝑈)
594, 58sylan2 586 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 384  w3o 1106  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  {crab 3059  cun 3730  wss 3732   cint 4633   class class class wbr 4809   Or wor 5197  dom cdm 5277  cres 5279  Oncon0 5908  suc csuc 5910  cfv 6068   No csur 32169   <s cslt 32170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-1o 7764  df-2o 7765  df-no 32172  df-slt 32173
This theorem is referenced by:  nosupbnd1lem1  32230  nosupbnd2  32238
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