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Theorem noresle 31971
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 3820 . . . 4 ((dom 𝑈𝐴 ∧ dom 𝑆𝐴) ↔ (dom 𝑈 ∪ dom 𝑆) ⊆ 𝐴)
2 ssralv 3699 . . . 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 207 . . 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 1280 . 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 4688 . . . . . . . 8 (𝑈 = 𝑆 → (𝑈 <s 𝑈𝑆 <s 𝑈))
65notbid 307 . . . . . . 7 (𝑈 = 𝑆 → (¬ 𝑈 <s 𝑈 ↔ ¬ 𝑆 <s 𝑈))
76biimpd 219 . . . . . 6 (𝑈 = 𝑆 → (¬ 𝑈 <s 𝑈 → ¬ 𝑆 <s 𝑈))
8 sltso 31952 . . . . . . . 8 <s Or No
9 sonr 5085 . . . . . . . 8 (( <s Or No 𝑈 No ) → ¬ 𝑈 <s 𝑈)
108, 9mpan 706 . . . . . . 7 (𝑈 No → ¬ 𝑈 <s 𝑈)
1110adantr 480 . . . . . 6 ((𝑈 No 𝑆 No ) → ¬ 𝑈 <s 𝑈)
127, 11impel 484 . . . . 5 ((𝑈 = 𝑆 ∧ (𝑈 No 𝑆 No )) → ¬ 𝑆 <s 𝑈)
1312adantrr 753 . . . 4 ((𝑈 = 𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ¬ 𝑆 <s 𝑈)
1413ex 449 . . 3 (𝑈 = 𝑆 → (((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)) → ¬ 𝑆 <s 𝑈))
15 simprl 809 . . . . 5 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → (𝑈 No 𝑆 No ))
16 simprll 819 . . . . . . . . . . 11 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑈 No )
17 simprlr 820 . . . . . . . . . . 11 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑆 No )
18 simpl 472 . . . . . . . . . . 11 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑈𝑆)
19 nosepne 31956 . . . . . . . . . . 11 ((𝑈 No 𝑆 No 𝑈𝑆) → (𝑈 {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ≠ (𝑆 {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}))
2016, 17, 18, 19syl3anc 1366 . . . . . . . . . 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 31943 . . . . . . . . . . . . 13 ((𝑈 No 𝑆 No 𝑈𝑆) → {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ On)
2216, 17, 18, 21syl3anc 1366 . . . . . . . . . . . 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 5841 . . . . . . . . . . . 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 6246 . . . . . . . . . 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 6246 . . . . . . . . . 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 2900 . . . . . . . . 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 2828 . . . . . . . 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 6228 . . . . . . . 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 135 . . . . . . 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 31959 . . . . . . . . 9 ((𝑈 No 𝑆 No 𝑈𝑆) → {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ (dom 𝑈 ∪ dom 𝑆))
3216, 17, 18, 31syl3anc 1366 . . . . . . . 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 811 . . . . . . . 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 5828 . . . . . . . . . . . 12 (𝑔 = {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} → suc 𝑔 = suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)})
3534reseq2d 5428 . . . . . . . . . . 11 (𝑔 = {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} → (𝑆 ↾ suc 𝑔) = (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}))
3634reseq2d 5428 . . . . . . . . . . 11 (𝑔 = {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} → (𝑈 ↾ suc 𝑔) = (𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}))
3735, 36breq12d 4698 . . . . . . . . . 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 307 . . . . . . . . 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 3336 . . . . . . . 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 7055 . . . . . . . . . 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 31938 . . . . . . . . 9 ((𝑈 No ∧ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ On) → (𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ∈ No )
4416, 42, 43syl2anc 694 . . . . . . . 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 31938 . . . . . . . . 9 ((𝑆 No ∧ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ On) → (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ∈ No )
4617, 42, 45syl2anc 694 . . . . . . . 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 5087 . . . . . . . . 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 706 . . . . . . . 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 694 . . . . . . 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 1476 . . . . . 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 31940 . . . . . . 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 1366 . . . . . 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 31783 . . . . . 6 (( <s Or No ∧ (𝑈 No 𝑆 No )) → (𝑈 <s 𝑆 → ¬ 𝑆 <s 𝑈))
558, 54mpan 706 . . . . 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 449 . . 3 (𝑈𝑆 → (((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)) → ¬ 𝑆 <s 𝑈))
5814, 57pm2.61ine 2906 . 2 (((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)) → ¬ 𝑆 <s 𝑈)
594, 58sylan2 490 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 383  w3o 1053  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  {crab 2945  cun 3605  wss 3607   cint 4507   class class class wbr 4685   Or wor 5063  dom cdm 5143  cres 5145  Oncon0 5761  suc csuc 5763  cfv 5926   No csur 31918   <s cslt 31919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-1o 7605  df-2o 7606  df-no 31921  df-slt 31922
This theorem is referenced by:  nosupbnd1lem1  31979  nosupbnd2  31987
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