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Theorem noextendgt 31948
 Description: Extending a surreal with a positive sign results in a bigger surreal. (Contributed by Scott Fenton, 22-Nov-2021.)
Assertion
Ref Expression
noextendgt (𝐴 No 𝐴 <s (𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩}))

Proof of Theorem noextendgt
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 nodmord 31931 . . . . . . . 8 (𝐴 No → Ord dom 𝐴)
2 ordirr 5779 . . . . . . . 8 (Ord dom 𝐴 → ¬ dom 𝐴 ∈ dom 𝐴)
31, 2syl 17 . . . . . . 7 (𝐴 No → ¬ dom 𝐴 ∈ dom 𝐴)
4 ndmfv 6256 . . . . . . 7 (¬ dom 𝐴 ∈ dom 𝐴 → (𝐴‘dom 𝐴) = ∅)
53, 4syl 17 . . . . . 6 (𝐴 No → (𝐴‘dom 𝐴) = ∅)
6 nofun 31927 . . . . . . . . 9 (𝐴 No → Fun 𝐴)
7 funfn 5956 . . . . . . . . 9 (Fun 𝐴𝐴 Fn dom 𝐴)
86, 7sylib 208 . . . . . . . 8 (𝐴 No 𝐴 Fn dom 𝐴)
9 nodmon 31928 . . . . . . . . 9 (𝐴 No → dom 𝐴 ∈ On)
10 2on 7613 . . . . . . . . 9 2𝑜 ∈ On
11 fnsng 5976 . . . . . . . . 9 ((dom 𝐴 ∈ On ∧ 2𝑜 ∈ On) → {⟨dom 𝐴, 2𝑜⟩} Fn {dom 𝐴})
129, 10, 11sylancl 695 . . . . . . . 8 (𝐴 No → {⟨dom 𝐴, 2𝑜⟩} Fn {dom 𝐴})
13 disjsn 4278 . . . . . . . . 9 ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ↔ ¬ dom 𝐴 ∈ dom 𝐴)
143, 13sylibr 224 . . . . . . . 8 (𝐴 No → (dom 𝐴 ∩ {dom 𝐴}) = ∅)
15 snidg 4239 . . . . . . . . 9 (dom 𝐴 ∈ On → dom 𝐴 ∈ {dom 𝐴})
169, 15syl 17 . . . . . . . 8 (𝐴 No → dom 𝐴 ∈ {dom 𝐴})
17 fvun2 6309 . . . . . . . 8 ((𝐴 Fn dom 𝐴 ∧ {⟨dom 𝐴, 2𝑜⟩} Fn {dom 𝐴} ∧ ((dom 𝐴 ∩ {dom 𝐴}) = ∅ ∧ dom 𝐴 ∈ {dom 𝐴})) → ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘dom 𝐴) = ({⟨dom 𝐴, 2𝑜⟩}‘dom 𝐴))
188, 12, 14, 16, 17syl112anc 1370 . . . . . . 7 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘dom 𝐴) = ({⟨dom 𝐴, 2𝑜⟩}‘dom 𝐴))
19 fvsng 6488 . . . . . . . 8 ((dom 𝐴 ∈ On ∧ 2𝑜 ∈ On) → ({⟨dom 𝐴, 2𝑜⟩}‘dom 𝐴) = 2𝑜)
209, 10, 19sylancl 695 . . . . . . 7 (𝐴 No → ({⟨dom 𝐴, 2𝑜⟩}‘dom 𝐴) = 2𝑜)
2118, 20eqtrd 2685 . . . . . 6 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘dom 𝐴) = 2𝑜)
225, 21jca 553 . . . . 5 (𝐴 No → ((𝐴‘dom 𝐴) = ∅ ∧ ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘dom 𝐴) = 2𝑜))
23223mix3d 1258 . . . 4 (𝐴 No → (((𝐴‘dom 𝐴) = 1𝑜 ∧ ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘dom 𝐴) = ∅) ∨ ((𝐴‘dom 𝐴) = 1𝑜 ∧ ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘dom 𝐴) = 2𝑜) ∨ ((𝐴‘dom 𝐴) = ∅ ∧ ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘dom 𝐴) = 2𝑜)))
24 fvex 6239 . . . . 5 (𝐴‘dom 𝐴) ∈ V
25 fvex 6239 . . . . 5 ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘dom 𝐴) ∈ V
2624, 25brtp 31765 . . . 4 ((𝐴‘dom 𝐴){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘dom 𝐴) ↔ (((𝐴‘dom 𝐴) = 1𝑜 ∧ ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘dom 𝐴) = ∅) ∨ ((𝐴‘dom 𝐴) = 1𝑜 ∧ ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘dom 𝐴) = 2𝑜) ∨ ((𝐴‘dom 𝐴) = ∅ ∧ ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘dom 𝐴) = 2𝑜)))
2723, 26sylibr 224 . . 3 (𝐴 No → (𝐴‘dom 𝐴){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘dom 𝐴))
2810elexi 3244 . . . . . 6 2𝑜 ∈ V
2928prid2 4330 . . . . 5 2𝑜 ∈ {1𝑜, 2𝑜}
3029noextenddif 31946 . . . 4 (𝐴 No {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘𝑥)} = dom 𝐴)
3130fveq2d 6233 . . 3 (𝐴 No → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘𝑥)}) = (𝐴‘dom 𝐴))
3230fveq2d 6233 . . 3 (𝐴 No → ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘𝑥)}) = ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘dom 𝐴))
3327, 31, 323brtr4d 4717 . 2 (𝐴 No → (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘𝑥)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘𝑥)}))
3429noextend 31944 . . 3 (𝐴 No → (𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩}) ∈ No )
35 sltval2 31934 . . 3 ((𝐴 No ∧ (𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩}) ∈ No ) → (𝐴 <s (𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩}) ↔ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘𝑥)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘𝑥)})))
3634, 35mpdan 703 . 2 (𝐴 No → (𝐴 <s (𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩}) ↔ (𝐴 {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘𝑥)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘ {𝑥 ∈ On ∣ (𝐴𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})‘𝑥)})))
3733, 36mpbird 247 1 (𝐴 No 𝐴 <s (𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩}))
 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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