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Theorem noextendlt 31947
 Description: Extending a surreal with a negative sign results in a smaller surreal. (Contributed by Scott Fenton, 22-Nov-2021.)
Assertion
Ref Expression
noextendlt (𝐴 No → (𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩}) <s 𝐴)

Proof of Theorem noextendlt
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 nofun 31927 . . . . . . . . 9 (𝐴 No → Fun 𝐴)
2 funfn 5956 . . . . . . . . 9 (Fun 𝐴𝐴 Fn dom 𝐴)
31, 2sylib 208 . . . . . . . 8 (𝐴 No 𝐴 Fn dom 𝐴)
4 nodmon 31928 . . . . . . . . 9 (𝐴 No → dom 𝐴 ∈ On)
5 1on 7612 . . . . . . . . 9 1𝑜 ∈ On
6 fnsng 5976 . . . . . . . . 9 ((dom 𝐴 ∈ On ∧ 1𝑜 ∈ On) → {⟨dom 𝐴, 1𝑜⟩} Fn {dom 𝐴})
74, 5, 6sylancl 695 . . . . . . . 8 (𝐴 No → {⟨dom 𝐴, 1𝑜⟩} Fn {dom 𝐴})
8 nodmord 31931 . . . . . . . . . 10 (𝐴 No → Ord dom 𝐴)
9 ordirr 5779 . . . . . . . . . 10 (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
108, 9syl 17 . . . . . . . . 9 (𝐴 No → ¬ dom 𝐴 ∈ dom 𝐴)
11 disjsn 4278 . . . . . . . . 9 ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
1210, 11sylibr 224 . . . . . . . 8 (𝐴 No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
13 snidg 4239 . . . . . . . . 9 (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
144, 13syl 17 . . . . . . . 8 (𝐴 No → dom 𝐴 ∈ {dom 𝐴})
15 fvun2 6309 . . . . . . . 8 ((𝐴 Fn dom 𝐴 ∧ {⟨dom 𝐴, 1𝑜⟩} Fn {dom 𝐴} ∧ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ∧ dom 𝐴 ∈ {dom 𝐴})) → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = ({⟨dom 𝐴, 1𝑜⟩}‘dom 𝐴))
163, 7, 12, 14, 15syl112anc 1370 . . . . . . 7 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = ({⟨dom 𝐴, 1𝑜⟩}‘dom 𝐴))
17 fvsng 6488 . . . . . . . 8 ((dom 𝐴 ∈ On ∧ 1𝑜 ∈ On) → ({⟨dom 𝐴, 1𝑜⟩}‘dom 𝐴) = 1𝑜)
184, 5, 17sylancl 695 . . . . . . 7 (𝐴 No → ({⟨dom 𝐴, 1𝑜⟩}‘dom 𝐴) = 1𝑜)
1916, 18eqtrd 2685 . . . . . 6 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = 1𝑜)
20 ndmfv 6256 . . . . . . 7 (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
2110, 20syl 17 . . . . . 6 (𝐴 No → (𝐴‘dom 𝐴) = ∅)
2219, 21jca 553 . . . . 5 (𝐴 No → (((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = 1𝑜 ∧ (𝐴‘dom 𝐴) = ∅))
23223mix1d 1256 . . . 4 (𝐴 No → ((((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = 1𝑜 ∧ (𝐴‘dom 𝐴) = ∅) ∨ (((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = 1𝑜 ∧ (𝐴‘dom 𝐴) = 2𝑜) ∨ (((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = ∅ ∧ (𝐴‘dom 𝐴) = 2𝑜)))
24 fvex 6239 . . . . 5 ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) ∈ V
25 fvex 6239 . . . . 5 (𝐴‘dom 𝐴) ∈ V
2624, 25brtp 31765 . . . 4 (((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴‘dom 𝐴) ↔ ((((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = 1𝑜 ∧ (𝐴‘dom 𝐴) = ∅) ∨ (((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = 1𝑜 ∧ (𝐴‘dom 𝐴) = 2𝑜) ∨ (((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴) = ∅ ∧ (𝐴‘dom 𝐴) = 2𝑜)))
2723, 26sylibr 224 . . 3 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴‘dom 𝐴))
28 necom 2876 . . . . . . 7 (((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥) ↔ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥))
2928rabbii 3216 . . . . . 6 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)} = {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥)}
3029inteqi 4511 . . . . 5 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)} = {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥)}
315elexi 3244 . . . . . . 7 1𝑜 ∈ V
3231prid1 4329 . . . . . 6 1𝑜 ∈ {1𝑜, 2𝑜}
3332noextenddif 31946 . . . . 5 (𝐴 No {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥)} = dom 𝐴)
3430, 33syl5eq 2697 . . . 4 (𝐴 No {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)} = dom 𝐴)
3534fveq2d 6233 . . 3 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)}) = ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘dom 𝐴))
3634fveq2d 6233 . . 3 (𝐴 No → (𝐴 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)}) = (𝐴‘dom 𝐴))
3727, 35, 363brtr4d 4717 . 2 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)}))
3832noextend 31944 . . 3 (𝐴 No → (𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩}) ∈ No )
39 sltval2 31934 . . 3 (((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩}) ∈ No 𝐴 No ) → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩}) <s 𝐴 ↔ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)})))
4038, 39mpancom 704 . 2 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩}) <s 𝐴 ↔ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩})‘𝑥) ≠ (𝐴𝑥)})))
4137, 40mpbird 247 1 (𝐴 No → (𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩}) <s 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∨ w3o 1053   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  {crab 2945   ∪ cun 3605   ∩ cin 3606  ∅c0 3948  {csn 4210  {ctp 4214  ⟨cop 4216  ∩ cint 4507   class class class wbr 4685  dom cdm 5143  Ord word 5760  Oncon0 5761  Fun wfun 5920   Fn wfn 5921  ‘cfv 5926  1𝑜c1o 7598  2𝑜c2o 7599   No csur 31918
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