MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  noresle Structured version   Visualization version   GIF version

Theorem noresle 27549
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 4177 . . . 4 ((dom 𝑈𝐴 ∧ dom 𝑆𝐴) ↔ (dom 𝑈 ∪ dom 𝑆) ⊆ 𝐴)
2 ssralv 4043 . . . 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 5142 . . . . . . . 8 (𝑈 = 𝑆 → (𝑈 <s 𝑈𝑆 <s 𝑈))
65notbid 318 . . . . . . 7 (𝑈 = 𝑆 → (¬ 𝑈 <s 𝑈 ↔ ¬ 𝑆 <s 𝑈))
76biimpd 228 . . . . . 6 (𝑈 = 𝑆 → (¬ 𝑈 <s 𝑈 → ¬ 𝑆 <s 𝑈))
8 sltso 27528 . . . . . . . 8 <s Or No
9 sonr 5602 . . . . . . . 8 (( <s Or No 𝑈 No ) → ¬ 𝑈 <s 𝑈)
108, 9mpan 687 . . . . . . 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 714 . . . 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 768 . . . . 5 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → (𝑈 No 𝑆 No ))
16 simprll 776 . . . . . . . . . . 11 ((𝑈𝑆 ∧ ((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → 𝑈 No )
17 simprlr 777 . . . . . . . . . . 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 27532 . . . . . . . . . . 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 27517 . . . . . . . . . . . . 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 6436 . . . . . . . . . . . 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 6902 . . . . . . . . . 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 6902 . . . . . . . . . 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 3010 . . . . . . . . 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 2937 . . . . . . . 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 6881 . . . . . . . 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 27536 . . . . . . . . 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 770 . . . . . . . 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 6421 . . . . . . . . . . . 12 (𝑔 = {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} → suc 𝑔 = suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)})
3534reseq2d 5972 . . . . . . . . . . 11 (𝑔 = {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} → (𝑆 ↾ suc 𝑔) = (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}))
3634reseq2d 5972 . . . . . . . . . . 11 (𝑔 = {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} → (𝑈 ↾ suc 𝑔) = (𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}))
3735, 36breq12d 5152 . . . . . . . . . 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 3600 . . . . . . . 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 7793 . . . . . . . . . 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 27512 . . . . . . . . 9 ((𝑈 No ∧ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ On) → (𝑈 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ∈ No )
4416, 42, 43syl2anc 583 . . . . . . . 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 27512 . . . . . . . . 9 ((𝑆 No ∧ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)} ∈ On) → (𝑆 ↾ suc {𝑥 ∈ On ∣ (𝑈𝑥) ≠ (𝑆𝑥)}) ∈ No )
4617, 42, 45syl2anc 583 . . . . . . . 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 5604 . . . . . . . . 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 687 . . . . . . . 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 583 . . . . . . 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 27514 . . . . . . 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 5610 . . . . . 6 (( <s Or No ∧ (𝑈 No 𝑆 No )) → (𝑈 <s 𝑆 → ¬ 𝑆 <s 𝑈))
558, 54mpan 687 . . . . 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 3017 . 2 (((𝑈 No 𝑆 No ) ∧ ∀𝑔 ∈ (dom 𝑈 ∪ dom 𝑆) ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔)) → ¬ 𝑆 <s 𝑈)
594, 58sylan2 592 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 1083  w3a 1084   = wceq 1533  wcel 2098  wne 2932  wral 3053  {crab 3424  cun 3939  wss 3941   cint 4941   class class class wbr 5139   Or wor 5578  dom cdm 5667  cres 5669  Oncon0 6355  suc csuc 6357  cfv 6534   No csur 27492   <s cslt 27493
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 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-tp 4626  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ord 6358  df-on 6359  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-1o 8462  df-2o 8463  df-no 27495  df-slt 27496
This theorem is referenced by:  nosupbnd1lem1  27560  nosupbnd2  27568  noinfbnd1lem1  27575  noinfbnd2  27583
  Copyright terms: Public domain W3C validator