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Theorem noextendlt 33060
 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 33040 . . . . . . . . 9 (𝐴 No → Fun 𝐴)
2 funfn 6382 . . . . . . . . 9 (Fun 𝐴𝐴 Fn dom 𝐴)
31, 2sylib 219 . . . . . . . 8 (𝐴 No 𝐴 Fn dom 𝐴)
4 nodmon 33041 . . . . . . . . 9 (𝐴 No → dom 𝐴 ∈ On)
5 1on 8100 . . . . . . . . 9 1o ∈ On
6 fnsng 6403 . . . . . . . . 9 ((dom 𝐴 ∈ On ∧ 1o ∈ On) → {⟨dom 𝐴, 1o⟩} Fn {dom 𝐴})
74, 5, 6sylancl 586 . . . . . . . 8 (𝐴 No → {⟨dom 𝐴, 1o⟩} Fn {dom 𝐴})
8 nodmord 33044 . . . . . . . . . 10 (𝐴 No → Ord dom 𝐴)
9 ordirr 6207 . . . . . . . . . 10 (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
108, 9syl 17 . . . . . . . . 9 (𝐴 No → ¬ dom 𝐴 ∈ dom 𝐴)
11 disjsn 4646 . . . . . . . . 9 ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
1210, 11sylibr 235 . . . . . . . 8 (𝐴 No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
13 snidg 4596 . . . . . . . . 9 (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
144, 13syl 17 . . . . . . . 8 (𝐴 No → dom 𝐴 ∈ {dom 𝐴})
15 fvun2 6752 . . . . . . . 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 1368 . . . . . . 7 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = ({⟨dom 𝐴, 1o⟩}‘dom 𝐴))
17 fvsng 6938 . . . . . . . 8 ((dom 𝐴 ∈ On ∧ 1o ∈ On) → ({⟨dom 𝐴, 1o⟩}‘dom 𝐴) = 1o)
184, 5, 17sylancl 586 . . . . . . 7 (𝐴 No → ({⟨dom 𝐴, 1o⟩}‘dom 𝐴) = 1o)
1916, 18eqtrd 2861 . . . . . 6 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = 1o)
20 ndmfv 6697 . . . . . . 7 (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
2110, 20syl 17 . . . . . 6 (𝐴 No → (𝐴‘dom 𝐴) = ∅)
2219, 21jca 512 . . . . 5 (𝐴 No → (((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) = 1o ∧ (𝐴‘dom 𝐴) = ∅))
23223mix1d 1330 . . . 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 6680 . . . . 5 ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴) ∈ V
25 fvex 6680 . . . . 5 (𝐴‘dom 𝐴) ∈ V
2624, 25brtp 32869 . . . 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 235 . . 3 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴‘dom 𝐴))
28 necom 3074 . . . . . . 7 (((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴𝑥) ↔ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥))
2928rabbii 3479 . . . . . 6 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴𝑥)} = {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥)}
3029inteqi 4878 . . . . 5 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴𝑥)} = {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥)}
31 1oex 8101 . . . . . . 7 1o ∈ V
3231prid1 4697 . . . . . 6 1o ∈ {1o, 2o}
3332noextenddif 33059 . . . . 5 (𝐴 No {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥)} = dom 𝐴)
3430, 33syl5eq 2873 . . . 4 (𝐴 No {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴𝑥)} = dom 𝐴)
3534fveq2d 6671 . . 3 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘ {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴𝑥)}) = ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘dom 𝐴))
3634fveq2d 6671 . . 3 (𝐴 No → (𝐴 {𝑥 ∈ On ∣ ((𝐴 ∪ {⟨dom 𝐴, 1o⟩})‘𝑥) ≠ (𝐴𝑥)}) = (𝐴‘dom 𝐴))
3727, 35, 363brtr4d 5095 . 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 33057 . . 3 (𝐴 No → (𝐴 ∪ {⟨dom 𝐴, 1o⟩}) ∈ No )
39 sltval2 33047 . . 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 684 . 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 258 1 (𝐴 No → (𝐴 ∪ {⟨dom 𝐴, 1o⟩}) <s 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ w3o 1080   = wceq 1530   ∈ wcel 2107   ≠ wne 3021  {crab 3147   ∪ cun 3938   ∩ cin 3939  ∅c0 4295  {csn 4564  {ctp 4568  ⟨cop 4570  ∩ cint 4874   class class class wbr 5063  dom cdm 5554  Ord word 6188  Oncon0 6189  Fun wfun 6346   Fn wfn 6347  ‘cfv 6352  1oc1o 8086  2oc2o 8087   No csur 33031
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